And in recent weeks, he has been reported to have stepped up his recovery process with training sessions with the U23s and the Toffees' main setup.
In his first pre-match briefing at the helms of the Goodison Park outfit - and ahead of Sunday's Merseyside derby against Liverpool - Allardyce stated that the 28-year-old could be near a return to action, given his healing process continues, uninterrupted.
"Bolasie has started training with us, which is good news," Allardyce said.
"A couple of games with the Under 23s or a behind closed doors game [could be all that he needs].
"As long as there is no reaction to the knee, we could see him on the subs bench to begin with."
Bolasie's endured an injury-hit debut season at Everton on the back of a big move from Crystal Palace. He was allowed just 13 top-flight appearances with a solitary strike to show forth.