The Gunners face Pep Guardiola's side at Wembley Stadium on February 25 following both legs of their Europa League last-32 tie against Swedish side Ostersunds.
Ramsey, however, did not train on Wednesday afternoon at London Colney and was not part of the squad who flew out to Sweden in the evening.
Wenger has now confirmed that the midfielder's recovery is not progressing as he had expected.
"It's a little bit a sensitive case," the Arsenal boss told reporters at his pre-match press conference. "I expect him to be a bit further in the development of the groin problem.
"He doesn't move forward as quickly as I expected. Today I would say certainly he will not be available for the second game against Ostersunds.
"Maybe for the cup final. I don't know yet, it depends how he will progress from now."
Ramsey has made 18 appearances for Arsenal this season, scoring six goals and registering eight assists.