Indeed they did. The most famous of the European envoys was Ruy González de Clavijo, ambassador of Henry III of Castile to Timur, whose detailed account, Narrative of the embassy of Ruy Gonzalez de Clavijo to the court of Timour at Samarcand, A.D. 1403-6, has survived: https://archive.org/details/narrativeembass00markgoog

Clavijo's visit was useful to Timur in multiple ways; beyond his diplomacy with Europe, it provided a convenient way for him to offend the Ming (by treating Clavijo as of higher rank than the Ming envoys).

There were ambassadors from Europe, and Timur sent ambassadors to Europe, at least to Byzantium, France and Castile. The earliest of Timur's ambassadors to Europe were to Byzantium, seeking support for his war against the Ottomans (or at least, to stop aid to the Ottomans). His stunning victory over the Ottomans, capturing the sultan Bayazid in the process, at Ankara, 1402, made Timur somewhat of a celebrity in Europe, and stimulated the flow of European ambassadors to his court. But there were contacts even from western Europe before then - Henry III of Castile had sent two ambassadors to Timur already, before Ankara (the same ambassadors also visited Bayazid).

For a good overview of this diplomacy, see Knobler, A. (1995). "The Rise of Tīmūr and Western Diplomatic Response, 1390–1405". Journal of the Royal Asiatic Society, 5(3), 341-349. doi:10.1017/S135618630000660X