Snodgrass left the London Stadium to join the Baggies on an 18-month deal earlier this month.
West Ham had initially refused to sanction his move until January 20.
But Allardyce wanted the winger to play against Wolves, a game the Baggies went on to win last Saturday, leading to the deal to be struck.
Premier League rules, however, prevent that type of agreement over permanent transfers.
Speaking ahead of his side’s defeat to the Hammers, Big Sam confirmed there was “an agreement between the clubs that this game he would not be allowed to play. We could only get the deal done with that.”
After the match, he was less forthcoming and said: “I can’t answer that. I’ll have to wait and see.
“Before I answer anything that might cause me or West Ham any trouble then I’ll wait and see what the Premier League say but I’ve got bigger things to worry about.”
West Ham boss David Moyes said: “I’m not sure it’s any of my business. I don’t think its anyone else’s. The Premier League have to do what they have to do.”
Jarrod Bowen and Michail Antonio made it four wins on the spin for David Moyes’ side and moved them up to seventh in the table.
It also means West Ham have won each of their opening four games of a calendar year for only the second time in their history.
Defeat for West Brom, meanwhile, leaves them 19th, five points from safety.