The 20-year-old Pole - who started the season as the Gunners' third choice - suffered a dislocated finger in Tuesday night's Champions League defeat to Barcelona at the Nou Camp when he was hurt saving Dani Alves' 16th-minute free-kick.
Former number one Manuel Almunia, 33, came on to make some eye-catching saves as the Catalan side ran out 3-1 winners to end Arsenal's European hopes for another season.
However, with Lukasz Fabianski already ruled out for the rest of the season after shoulder surgery and Vito Mannone troubled by a thigh injury while on loan at Hull, Wenger looks set to ask the Premier League for permission to bring in experienced cover, with youngsters James Shea, 19, and Damian Martinez, 18, the only other fit keepers.
"We need now an experienced goalkeeper as cover for Almunia because Mannone is out for six weeks so we cannot call on him," Wenger told Arsenal TV Online.
"Mannone is here at the training ground having treatment, but he cannot play for the next six weeks.
"We can call him back [from Hull], but he is not available.
"We have Wojciech out - in my opinion for six weeks. We have Fabianski out for the rest of the season. We absolutely need some cover in goal."
Almunia had looked set to leave Arsenal during the January transfer window, but the Spaniard now finds himself thrust back into the first-team picture ahead of Saturday's FA Cup quarter-final at Manchester United.
Wenger has been impressed with Almunia's approach to the situation.
"He is very professional, [has a] very dignified attitude and overall he has done very well," the Arsenal boss said.
"Yes, he has done quite well when he came on [against Barcelona] and was quite busy in the final part of the game and he did very well."
Wenger does not expect any of his senior squad to be happy just warming the bench.
"When I took Jens Lehmann out, who [had] made only one mistake, and put Almunia in, Lehmann came in the next day and fought even harder," said the Frenchman. "That's what I believe our job is about."