The Hammers have failed to score in six of their 10 Premier League games so far this season, with Allardyce reverting to a system without a recognised centre-forward in three of those matches.
With summer signing Andy Carroll still on the treatment table Allardyce is hoping to add quality in the final third when the January transfer window opens, revealing that only a big impact from either Carlton Cole and Mladen Petric will change that stance.
He told The Mirror: "We tried to get three top loan strikers who play in this country but we failed to get all three. They went elsewhere and played for another club and one stayed where he was.
"We will have to try again depending how Carlton and Mladen go before the end of January.
"If they come good and Andy gets back there will be no need for another striker."
Ba is thought to be on West Ham's list of transfer targets following a failed loan move last summer, with Oscar Cardozo and Alfred Finnbogason also options being considered by the Clarets in a bid to beef up their forward line.
West Ham's latest league game saw both sides fail to register last weekend, with Aston Villa playing out a goalless draw against the Hammers at Upton Park.