The former Liverpool frontman has not featured this season since aggravating an Achilles injury before the 2013-14 campaign.
Following various treatment with medical professionals overseas, Allardyce has claimed that the ex-Newcastle United player is edging closer to fitness.
"We hope Andy will be back with us to start some training next week," Allardyce said in a press conference. "It's tentative in our approach to it. We can't get over-anxious and push him too hard. We need to get him fit to stay fit."
The Hammers are only one place above the bottom three in the Premier League table on goal difference.