Winterburn believes the Gunners will miss out on the top four this season and also lose long-serving midfielder Ramsey.
"As Arsenal rebuild and restructure the team, they have to stay around the top four and be in a position to challenge the teams above them, but I think this season has come too soon for the club.
He added: "I think Ramsey isn't committing to a new Arsenal contract because he wants to see where they finish this season.
"With a new manager coming in, there's a lot of uncertainty around the club and I would probably do the same.
"I think Ramsey loves the club, but he is looking around the dressing room and he isn't convinced this group of players can qualify for major honours in the near future."