The Bhoys failed to get past the qualifying rounds last season, and the former Celtic striker suggests that they could endure the same disappointment if they struggle at the back.
“Recruitment is a big thing,” the 42-year-old told BT Sport in the build-up to the Scottish Premiership clash against St. Johnstone away from home on Friday.
“Defensively Celtic have come up short in Europe. I think in the domestic league, they have conceded 17 goals in 36 games, which is good, but in Europe they have struggled defensively.
“They need to address that. For me, they need a left-back. They have got centre-half issues. Will (Virgil) van Dijk go, will he stay? (Jason) Denayer is off.”
Just like last season, Scottish Premiership champions Celtic will have to to negotiate tough qualifying rounds.
Last season, after getting the better of KR Reykjavik and Legia Warsaw in the second and third qualifying rounds respectively, Celtic faced NK Maribor in the playoff round.
Although the Scottish giants drew the first leg of the tie 1-1 in Slovenia, they lost the second leg 1-0 at Celtic Park.
As Sutton suggests, Celtic will have to be defensively much better in Europe than they were last term.
One suspects that manager Ronny Deila knows that already, and fans can expect the Norwegian to make necessary changes to make sure that the Hoops are competent in the Champions League qualifiers.