The Argentina international suffered the injury during Chelsea's 2-0 Premier League victory over Everton last weekend following a collision with defender Jarrad Branthwaite. He was taken to hospital for treatment.
"He's been better; he had the surgery. We still need to wait. (He will be out for) a few weeks. Hopefully less than more," Alonso said on Friday.
Mike Penders is expected to start in goal when Chelsea host Tottenham Hotspur on Saturday, with Martinez facing a spell on the sidelines.
Alonso expressed confidence in the young goalkeeper, pointing to his performances in cup matches and his brief appearance against Everton.
"He is ready," the Chelsea manager said. "He has been training well and played a couple of cup games where he showed he has a good level."
Penders also impressed Alonso when he came on against Everton, with the Spaniard highlighting his composure and ability.
"When he came on against Everton, he showed in a few minutes his personality and qualities," Alonso added.
The Chelsea boss also provided updates on Malo Gusto and Pep Chavarria. Gusto has recovered and is available, but Chavarria remains short of fitness despite increasing his training workload.
"Malo is fine. Pep is training a little bit more but he is not going to make it," Alonso said.
Bradley Barcola is available for Liverpool's Premier League meeting with Brighton at Anfield on Sunday, but Dominik Szoboszlai has been ruled out as the Reds continue a demanding run of fixtures.
Arsenal manager Mikel Arteta could hand Max Dowman an increased role in the first team after Bukayo Saka was ruled out for up to eight weeks with a quad injury.
Manchester United head coach Michael Carrick will be looking for an immediate response when his side host Bournemouth at Old Trafford, after their humiliating Champions League defeat to Como.
Oliver Glasner has praised Nottingham Forest's determination after they were held to a 1-1 draw by Ipswich Town, insisting his team had created enough chances to take all three points.
