Tevez has not featured at all for Manchester City since his falling out with manager Roberto Mancini back in September.
According to the Mirror, Liverpool boss Kenny Dalglish is hoping City will allow the Argentine to leave on loan before the transfer window shuts.
West Ham however are hoping to wait until after the closure of the window and re-sign the man they brought from Corinthians in 2006, with the loan window in the Football League remaining open.
Tevez is still desperate for a move to AC Milan, but the deal fell through after the Italian giants were unwilling to match his £200,000-a-week wages.
Liverpool had reportedly offered to take Carlos Tevez from City, with Andy Carroll going the other way.
The bid was rejected by City, believing they could get a better deal for him in the summer.