Tevez returned to training at Manchester City on Thursday following his suspension for apparently refusing to go on as a substitute against Bayern Munich a fortnight ago.
The controversial striker's future at Eastlands remains uncertain, while West Ham have already had an initial enquiry about a possible loan move knocked back.
But Allardyce is monitoring the situation and has not given up hope of Tevez returning to the club he originally joined in 2006 along with fellow Argentinian Javier Mascherano.
When asked today if their was any possibility of landing Tevez, Allardyce said at his pre-match press conference: "The answer is I don't know.
"I suppose the situation, from my point of view, is just if and when the decision is finally made regarding what's happening with Carlos at Manchester City, then perhaps my co-chairman David Sullivan might want to pursue it.
"That would be with my blessing, obviously. [Sullivan] was the man who talked about it with me first and if we could get him back here, based on what he did here last time and if he did love the club as much as he did, and if it was a possibility, then I think it's an outside chance.
"We'll wait and see."
Tevez was a huge favourite during his one season at West Ham, scoring seven goals including the strike which kept them up at Manchester United on the final day of the campaign.
He was named the club's player of the year and the first time he returned to Upton Park as a United player, he saluted the crowd by crossing his arms in a 'Hammers' gesture.
Allardyce has also confirmed Tottenham loanee David Bentley is unlikely to play for West Ham again this season after he underwent knee surgery.
Source: Team Talk
Source: Team Talk