Argentinian striker Tevez has been suspended by Manchester City and told he will never play for them again after his apparent refusal to go on as a substitute at Bayern Munich on Tuesday.
West Ham's initial enquiry about a loan deal for the player who saved them from relegation in 2007 was given short shrift this week.
But Allardyce has insisted he would love to sign Tevez and claimed his alleged disruptive influence holds no fears.
"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 the Hammers boss.
"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?
"I've had worse than him. I've had some bad ones in my time. Just keep playing him, that's the answer."