Cfrgtkky

The Zimmermann telegram to the embassy in Mexico outright suggests that German Empire is trying to court Mexico as an ally in the Great War:

We intend to begin on the first of February unrestricted submarine warfare. We shall endeavor in spite of this to keep the United States of America neutral. In the event of this not succeeding, we make Mexico a proposal of alliance on the following basis: make war together, make peace together, generous financial support and an understanding on our part that Mexico is to reconquer the lost territory in Texas, New Mexico, and Arizona. The settlement in detail is left to you. You will inform the President of the above most secretly as soon as the outbreak of war with the United States of America is certain and add the suggestion that he should, on his own initiative, invite Japan to immediate adherence and at the same time mediate between Japan and ourselves. Please call the President's attention to the fact that the ruthless employment of our submarines now offers the prospect of compelling England in a few months to make peace.
Signed, ZIMMERMANN

Were there other countries that the Central Powers approached outright in the hope of an alliance? Maybe Spain or Sweden in the Europe? Countries in South America?

Just a note: I'm asking about approached independent at that time countries, not promising independence to various cultural regions under Allies occupation (i.e. Russian-occupied Poland and Ukraine).

_{(Presumably the question rules out the many co-belligerents of the Central Powers, for which Wikipedia has a detailed article.)}

The Kingdom of Romania is possibly the best example. In fact, the Romanians technically were allied to the Central Powers - they were a secret party to the Triple Alliance. At the start of the Great War, King Carol I wanted to honour his treaty ties, but his government followed the Italian lead and declared neutrality instead.

Subsequently, there was some diplomatic activity aimed at enticing Romanian entry on the Austro-German side:

Marghiloman attempted to bring about a revival of the alliance with the Central Powers, but on different premises. Romania should receive Bucovina as a compensation for her intervention on the side of Austria-Hungary, and Transylvania should be either given to Romania, or at least should be made independent from Hungary.

_{Szász, Zoltán. "The Transylvanian Question: Romania and the Belligerents, July-October 1914". Journal of Central European Affairs, vol 13.2 (1953).}

However, the Central Powers did not want Romania as an ally bad enough to offer up their own land as payment:

The Central Powers offered Romania Bessarabia, but none of Transylvania. The Hungarians, led by their premier, Count Istvan Tisza, had a great deal of influence in Vienna and were naturally reluctant to cede any Hungarian territory . . . he [believed] that the Romanians were opportunistic aggressors who, therefore, would be influenced by the course of the war rather than by territorial concessions.

Ultimately, the Romanian Prime Minister Ionel Brătianu sought to be on the winning side, and the winds never blew Germany's way long enough for him to overlook the deficient territorial offers. Eventually Romania defected to the Triple Entente over the promise of taking Transylvania from Austria-Hungary, though they turned out to have been too clever by half and the premature declaration ended in catastrophe.

Sweden would be another top contender. Though there was no treaty ties here, strong pro-German sympathies among Swedish elites, coupled with historical Russo-Swedish animosity, made Sweden an "obvious" potential ally for the Central Powers from the start. In fact, the Swedish government even leveraged this perception to their benefit in negotiations with the Entente:

[B]ecause the traditional ruling circles were in sympathy with the Central Powers, the government made a major political issue of the protection of neutral trading interests . . . In the face of Allied economic pressures, the Swedish government could bank on the fact that an appreciable proportion of Allied shipments to Russia passed through Swedish territory. Hence the Russians repeatedly urged caution upon their allies, fearing that trans-shipments might be held up and perhaps even that Sweden might ultimately enter the war on the side of the Central Powers.

_{Hardach, Gerd. The First World War, 1914–1918. University of California Press, 1981.}

By 1915, encouraged by signs such as Sweden's attempt to prevent the Italian entry on the side of the Entente, the Central Powers began to make overtures towards Sweden, led by none other than Zimmermann of the Mexican telegram.

Attention shifted to Sweden partly as a result of ill-considered Swedish diplomatic initiatives . . . On each occasion Sweden seemed to be less an impartial go-between than an agent of the Central Powers . . .

The first moves were made by Arthur Zimmermann [who offered] the prospect of Swedish leadership of a future Nordic bloc.

_{Salmon, Patrick. Scandinavia and the Great Powers, 1890–1940. Cambridge University Press, 2002.}

Nonetheless, unlike Italy or Romania, Sweden ultimately adhered to their declaration of neutrality and stayed out of the war.

There were several other prospects such as Persia, Ethiopia, or Greece, but they were neither particularly realistic nor all that potentially helpful.

'Naturally' Germany, Austria-Hungary and Turkey would be interested in anyone joining their cause.

This is not about who ended up in 1918 being at war with Germany, Austria-Hungary, Bulgaria and Ottomans. This is about all those considered by one side or the other as potentially siding with "them" from 1914–1918. And this list of attempts is surprisingly long as is the list of successes in that direction short.

Besides Mexico, this includes Italy, Switzerland, Norway, Netherlands, Luxemburg, Romania, Greece, Venezuela, Afghanistan, Abyssinia, Sweden, Japan.

Given the German initiatives in the far East, the "offering" of German colonial possessions might also indicate that the United States would have to be included (however theoretical and unsuccessful that turned out). That view of dreadful potentialities was initially shared by the British, as they were quite keen to restrict some of their naval blockade tactics so as not to anger the Americans, who preferred trading with all sides involved at the beginning.

The German plans and ambitions were fulfilled with Bulgaria's entry into he Central Powers alliance.

Wooing Romania was a constant throughout the war. Not really successful.

Hampered by tactical considerations potential allies like Sweden, Netherlands or Switzerland were seen as much more useful if they stayed 'neutral' so as to continue trade. In the case of Switzerland the neutrality was at times called into question: with the Grimm–Hoffmann affair. Sweden was equally sympathising with Germany, for the most part, and the busy man Zimmerman did offer a few perks as well.

In the wider picture of the globe German hopes were directed at Mexico, Venzuela in South-America, at Abyssinia and Afghanistan in Africa and Asia.

In the height of colonialism, not really many independent countries were left on the map, that did not clearly see more strategic value in staying neutral or siding with the Entente.

Although not explicitly independent countries for this purpose it just has to be noted that just like all those nationalities in the Russian Empire (as excluded from the question) every Muslim was considered a potential ally and the Germans tried to explicitly call them to Jihad against their Western oppressors. This hoped-for alliance from Morocco to Pakistan would then have to be expanded to India, as the Hindu–German Conspiracy falls into a similar league.

Perhaps tangential to the aim of the question: but with Japan Germany really hoped to come to an early end of official hostilities and negotiated in earnest what was mere tactics for "offering peace" in Europe.

As this was not "promised independence" one might also consider the fact that the Ottoman Empire engaged in a frenzy of diplomatic activity, mainly after the Russian Revolution and attempted to gain allies against the British and Russians.

And depending on viewing angles, the Germans also offered Greece assistance against the allies forcing them into their camp, albeit with some imperial Prussian elegance and decorum:

On the Central Powers' side, on 29 November 1915 Falkenhayn had publicly threatened that if Greece could not neutralize the Allied and Serbian forces in its soil, the Germans and their allies would cross the border and do it for them, and on 10 December, the German Foreign Ministry reacted to the new agreement between Greece and the Allies regarding their armies in Macedonia by demanding the same rights of free movement in Greek territory. To these demands, the Greek government answered on 22 December that it would not actively oppose a Central Powers invasion of its territory, provided that the Bulgarians did not participate, or at least stayed out of the cities, and the command of the operations was in German hands; that Bulgaria issued no territorial demands, and that the Central Powers forces would withdraw once their objectives were met; and that the Greek authorities remain in place.

_{- Holger H. Herwig: "German Visions of Empire in Venezela 1871–1914", Princeton University Press: Princeton, 1986.}