The relegation-threatened Hammers are short of attacking options due to Andy Carroll's long-term injury-enforced absence and Allardyce has said he will look for "one or two" additions in the new year.
Representatives of Gyan, who left Sunderland in 2012 to join Al Ain in the United Arab Emirates, have hinted the east Londoners are keen on taking the Ghana international to Upton Park on loan.
And Allardyce has admitted he is a player on the radar.
"We'll be linked with anybody that's a centre forward that might be coming back into this country," said Allardyce.
"So 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."
Asked if would not rule out a move but could not confirm an approach, Allardyce said: "Absolutely."
Captain Kevin Nolan will return after suspension for West Ham's home clash with Arsenal.
And Allardyce said his pivotal leader must now start weighing in with goals.
He said: "We've left ourselves with a massive programme by Christmas, and going into New Year with Premier League games, FA Cup games, a semi-final in the Capital One Cup, so we're going to be using the squad to its full extent.
"So the quicker I can get players back fit in January and find one or two additions as well, the better it will be for us.
"The most important thing we miss with Kevin is his goals, he's only scored one.
"We have to accept the criticism, take it on the chin and focus on how well we know we can play, and try to get back to the same level of performance - but when you're in command of a game, make sure you put the ball in the back of the net.
"Otherwise you end up where we are.
"We are where we are for one reason, because we haven't scored enough goals."
Allardyce has challenged his West Ham side to copy Chelsea, who shackled Arsenal at the Emirates Stadium on Monday, holding Arsene Wenger's men to a 0-0 draw.
He said: "The fact Arsenal went an entire 90 minutes at home and only sneaked one shot on target I think shows you how well Chelsea defended all night, but maybe the weather had something to do with that.
"It's something we could do with doing against them on Boxing Day, blocking them out like Chelsea did and seeing what we can get from there."
Defender James Tomkins remains central to Allardyce's Christmas plans, despite his arrest on Sunday.
The 24-year-old was charged with one count of assaulting a police officer, one count of resisting arrest and one count of being drunk and disorderly in a public place after an incident outside the Sugar Hut Village nightclub in Brentwood.
Allardyce said: "He's a footballer we need and is a big player for West Ham, he's West Ham born and bred."