Tevez had a spell with West Ham in 2007 and Allardyce has revealed that the Championship club would welcome a return.
"If he would like to join us and the chairman would like to pay for him then yes, why wouldn't you want a top-class player like that?" said Allardyce.
"I've worked with some world-class players in my time, they win you games when you least expect it.
"With a goal record like his why wouldn't you want him?"
The Hammers drew 2-2 with Crystal Palace this afternoon.