Allen did not rule Carr out for Sunday's game against the Tampa Bay Buccaneers, but the team will be cautious with the quarterback and it would be a surprise if he played next week, sources told ESPN's Adam Schefter.
Jameis Winston would face Tampa Bay, his former team, if Carr is unable to go.
Allen said that Carr felt better Monday morning and that he'll continue to be evaluated throughout the week.
"We're not making any decisions today; we're not ruling anything out," Allen said. "We'll see where he's at as the week goes on."
Allen did not specify the grade of the sprain.
"All I can tell you is that he was much better today than he was yesterday," Allen said. "We're going to continue to evaluate him, and when he's healthy enough that he can go out and perform and do the things that he needs to do to give our team a chance to win, then he'll be back in there."
Carr's playing status will likely be determined by his pain tolerance and ability to function, but Allen said the injury won't be made worse by him playing through it.
"I don't think it's a further risk of injury," Allen said. "It's 'can you function and do your job?'"
Carr was hurt in the third quarter of the Saints' 18-17 loss to the Green Bay Packers at Lambeau Field on Sunday. The Saints were up 17-0 when he took a sack that caused him to land hard on his shoulder.
He lay on the turf for several minutes as athletic trainers tended to him. He then went to the medical tent on the sideline briefly before leaving the game. Carr went to a local hospital for X-rays but flew back with the team Sunday.
Winston took over in Carr's absence, completing 10 of 16 passes for 101 yards. He was able to drive the team 47 yards to the Green Bay 34 in the last three minutes of the game, but Saints kicker Blake Grupe missed a potential game-winning field goal.
"I don't think what we were doing really changed at all," Allen said. "I thought Jameis did some good things in the game. We drove ourselves down there and gave ourselves an opportunity and didn't finish."
After the game, Allen said the Saints need to protect better, but he did not attribute the injury to the struggling offensive line.
"I think that was the one play I felt like he kind of hung on to [the ball] too long," Allen said.