Stekelenburg has been out since October with a groin injury, and has made just three appearances this season.
The 35-year-old is on the verge of a return to Everton training though, and Allardyce is grateful that his injury problems are lessening.
“Maarten Stekelenburg is nearly joining in training,” he said. “When I arrived, there were some long-term injuries and we seem to have recovered from them.”
Everton have recently welcomed back Yannick Bolasie and Seamus Coleman to their first-team, after both suffered with serious injury last season.
The Toffees are also waiting for Leighton Baines to return, as the left-back should help to solidify their defence.
Stekelenburg is likely to only be used in a reserve role, when he does make his comeback.
Jordan Pickford has been in impressive form for Everton this season, and the England international will surely continue as Allardyce’s side’s number one, despite the increased competition.