In English and Spanish, the words for welcome have an uncanny relation: the translation is almost completely (if not completely) literal.Bien means well and venidos means come/came in the plural or something along those lines. I looked up if it came from Latin (in English welcome), and it said that it didn’t. So, I am wondering if there is a reconstructed PIE phrase of invitation that translated as “(you’re) well, come” or something like that.
It seems that English welcome derives from wilcuma, where wil- seems to derives from proto-Germanic * wiljaną. On the other hand, Spanish bienvenido(s) (French bienvenue(s), Portuguese bem-vindo(s), ...) and similar expressions in other Romance language, are recent constructions, not attested in classical Latin. This seems to exclude a common origin of the English form and Romance forms.
The etymology of the two expressions goes back to different sources, but the conceptual link is somewhat similar. "Welcome" goes back to "will" and not to "well" (etymonline.com/word/welcome). So the semantic origin would be something along the lines of "your coming suits my will/wish". "Bienvenidos" and Romance cognates, instead, go back to Latin "bene" ('well'). There is no exhaustive explanation in the etymological dictionaries of Romance languages that I have consulted, but the semantics should be something like "I consider your coming good".
Accepting that "wiktionary suggests the Latin is a calque of the Frankish reflex [...] which can't be ruled out" (@Tristan), the question is about the origin of welcome.
The reconstructed root of come is also found as exponent in Sanskrit (eg. अध्वगत् (adhvagat, “traveller”) अवगन्तोस् (ávagantos, “to descend; to approach; to visit; to obtain; to undertake”), en.wiktionary: गम्).
In general, zero-grade affixes from verb stems are a common indo-euopean word formation mechanism, like credo and hair do (which is to some extend coincidence). As a noun or adjective, a noun stem would remain unchanged with a thematic vowel added, though many different derivational suffixes exist (eg. perfective reduplication जगाम (jagā́ma) < *gʷegʷóme, en.wiktionary).
So, welcome is rather sugggestive of an older typus of word formation. However, as an interjection (Welcome!, Happy Birthday!), the context is so erroded, it is very unlikely to yield any significant evidence.
before 900; Middle English < Scandinavian; compare Old Norse velkominn, equivalent to velwell1 + kominncome (past participle); replacing Old English wilcuma one who is welcome, equivalent to wil- welcome (see will2) + cuma comer
Ideally, it should tell us something new about the root *gʷem- and its derivation.