The Wales international scored a hat-trick in the 5-1 Premier League thrashing of Everton at the start of the month but was forced to miss the 1-0 defeat to Tottenham due to a groin problem.
Ramsey did not travel with the squad for the first leg of the UEFA Europa League last-32 tie with Ostersunds this week and Wenger also expects him to miss the return leg at Emirates Stadium.
Wenger is concerned at the way the 27-year-old's recovery has progressed and fears he could be unavailable for the Wembley showdown with City on 26 February.
"Ramsey is a little bit of a sensitive case," he said. "He doesn't move forward as quickly as I expected.
"I would say certainly he will not be available for the second game against Ostersunds. For the cup final, I don't know. It depends on how he progresses from now."
Ramsey joins Alexandre Lacazette, Jack Wilshere, Laurent Koscielny and Petr Cech in sitting out the game at Jamtkraft Arena this week.
Lacazette is expected to miss up to six weeks after undergoing knee surgery, while Wilshere and Koscielny are being given extra time to recover from more minor complaints.
"Of course, you have heard about Lacazette," said Wenger. "Apart from that, Wilshere, Koscielny, Cech have stayed behind. Everyone else has travelled.
"One or two had recent problems. Koscielny with his Achilles, Wilshere as well.
"I wanted to rest Jack because he was sick in the game against Spurs and felt he hadn't fully recovered."