The 59-year-old admits the club have enquired about the Ghanaian, while also commented on James Tomkins, who was charged with assaulting a police officer
The Ghana international has been linked with a loan move to Upton Park as Allardyce attempts to add some firepower to his struggling side, who have scored just 14 goals in 17 Premier League matches.
With Andy Carroll having yet to feature following his move from Liverpool, Allardyce has made no secret of his desire to bolster his strikeforce and admits former Sunderland striker Gyan could be a target.
"We'll be linked with anybody that's a centre-forward that might be coming back into the country," said the West Ham boss ahead of the side's Boxing Day clash with Arsenal.
"If there's a player of any sort of quality we try and find out what the situation might be - but it is nothing more than that."
Allardyce was also questioned about defender James Tomkins, who had been charged with assaulting a police officer, and revealed that the 24-year-old has not broken any club rules.
"From a legal point of view it's difficult for me to comment," Allardyce continued.
"The only thing I can say is that I don't think James should have been there irrespective of whether he got into trouble or not.
"It's probably the highest profile public house in this country and I think it would have been better if he'd have stayed away.
"From a club point of view there's nothing broken in terms of rules because it was five or six days ahead of the next game. I can't comment any further than that."