The Welsh midfielder had to be taken off just 19 minutes after coming on as a substitute in the FA Cup fifth-round replay win over Hull City on Tuesday when he picked up a thigh injury.
Reports in recent days suggested that the 25-year-old could miss the rest of the season after suffering muscle damage, and that his participation in Wales' first major tournament since 1958 could even be in doubt.
Wenger has allayed those concerns, but admits that the problem is more serious than he had initially hoped and means Ramsey could miss crucial games during the run-in in the final weeks of the season.
The French manager told reporters, as quoted by the Evening Standard: "I don't think that he will miss the remainder of the season. At the moment his clinical signs are quite positive, at the start I would say four weeks.
"It's a bit worse than I thought at first. Usually when a player gets injured he stops, I thought it was just a precaution and he would be available for Sunday."
The news will come as a blow for Gunners fans, with Ramsey having been a key member of Wenger's side this season, scoring six goals in 33 appearances in all competitions from midfield.
Meanwhile, Wenger also gave an update on the fitness of central defensive trio Laurent Koscielny, Per Mertesacker, and Gabriel Paulista.
He added: "From the Hull game we lost Ramsey, Gabriel came off injured, Mertesacker injured. They should be available. Koscielny is not far, will I risk him? I don't know, I haven't made a decision.
"Laurent's got to join in normal training tomorrow and then we'll think. I'll think after advice from the medical team. If he's available I'll consider playing him."
After their midweek replay, Arsenal are straight back into FA Cup action this weekend, hosting Watford in the quarter-final stage at the Emirates Stadium on Sunday.