Celtic face Lincoln Red Imps in the second leg of their Champions League qualifier.
New Celtic manager Brendan Rodgers is presented to the fans after the press conference
Ambrose was one of the starters a week ago as Celtic lost to Lincoln Red Imps, and Sutton believes his time at the club has run out.
Celtic's Efe Ambrose looks dejected at the end of the match
Speaking on 5 Live's Monday Night Football Club, Sutton said that if Rodgers keeps on picking Ambrose, he risks driving a wedge between himself and supporters.
He said: "Brendan Rodgers has a bit of a problem, with his team selection if he keeps on picking Efe Ambrose.
"He will cause himself a big problem with his support if he keeps picking Efe Ambrose. He has cost Celtic in games, like Ross County last year. He is an accident waiting to happen."
Cetic's Efe Ambrose in action with Dundee United's Ryan Dow
Sutton pointed to Mikael Lustig as a player he would rather see at centre-half than Ambrose, who was on the bench against Red Imps.
Ultimately the former Celtic striker suggested the answer lies in the transfer market, and that Rodgers will not find it difficult to bring in a replacement.
He added: "Finding a centre-half better than Efe Ambrose, believe me, isn't very difficult."
Celtic may well be looking to sign another defender, and those plans could be dependent on the financial boost Champions League qualification will bring, which makes the second leg of their tie all the more important.
New Celtic manager Brendan Rodgers during the press conference