The winger has been sidelined for over a year with a cruciate knee ligament injury but is closing in on a comeback and may feature when Swansea visit Goodison Park on Sunday.
Allardyce, who took charge of Everton at the end of last month, cannot wait to have the former Crystal Palace man available for selection and he is convinced Bolasie's return will provide the team with a huge boost.
"The team has been crying out for what he offers," the manager 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."
However, Allardyce insists he won't rush the 28-year-old back into action too soon and plans to ease Bolasie slowly back into the side to avoid the risk of further injury.
He added: "It will be very pleasing to have him back in the squad but I will be very delicate with the introduction. At the very best it will be from the subs' bench and the minutes he gets will have to be the right thing for him and the team as well, depending on how the game is going. We can't expect too much. He will be a little rusty but if he does hit his level straight away that would be absolutely fantastic. He's a big player to have back.
"Having spoken to Yannick I know he's desperate to get back but we have to control that desire. It doesn't matter how he thinks he feels, it's about making sure we do the right process so that he's back for a long period of time and he doesn't injure that knee again."
Bolasie made two appearances for Everton's under-23s last week.