The 28-year-old has been missing for over a year with cruciate knee ligament damage, restricting him to just 15 first-team outings since arriving from Crystal Palace 16 months ago.
Toffees boss Allardyce hopes to have Bolasie available for the meeting with Swansea at Goodison Park, admitting that his side are desperate for an injection of pace and trickery in wide areas.
"The team has been crying out for what he offers," he told reporters. "Yannick is blessed with dynamic movements and pace that cause defenders great concern. He probably occupies two players when he's at his best because he's such a danger in possession.
"He's a player who has come from the lower leagues and built his way up to the top level of football. I always think those players have a great desire to achieve that. In the early part they've had people telling them they're not good enough but their own drive, determination and belief is so strong they go on and, in Yannick's case, arrive at Everton football club.
"It's been a fantastic journey for him and because of that there's a great opportunity for him to stay at the top level for as long as possible. Hopefully he will make a big contribution to our team when he comes back."
Congo international Bolasie has played in a couple of Under-23s matches over the past week to help build his fitness.