The Parkhead defender suffered the injury in Saturday’s 2-1 Ladbrokes Premiership defeat by Aberdeen at Pittodrie.
Speaking at Glasgow airport before the Celtic squad flew out to Amsterdam, former Parkhead midfielder Collins said: “Charlie is the only one not here. He has a thigh muscle injury, so we’re still waiting on confirmation of the scan result but it looks like it will be a few weeks.
“It is a blow, he’s an experienced player. He’s played in a lot of big games so we just hope he has a speedy recovery.”
Collins admitted Mulgrew’s absence means there is a good chance Croatia Under-21 defender Jozo Simunovic, signed from Dinamo Zagreb for a fee reported to be as much as £5.5million, will make his debut in the Amsterdam ArenA.
He said: “The team has not been picked but Jozo trained very well and there’s every chance he could get his debut.
“It’s the start of a group and it’s going to be a tough challenge.
“Ajax are a good young team. I watched their Champions League qualifiers and they were very unlucky not to qualify.
“They are a great club with a great history, so it’s going to be a wonderful challenge for us.
“We are looking for a reaction (from the Aberdeen game), a good performance, but we are going into a different environment and it’s going to be a great challenge for the players, a new challenge.
“We don’t play teams like Ajax every week, they have a lot of new players, doing very well and only lost their first points at the weekend in a 2-2 draw.
“It is another game, another tournament, and I thought we did fairly well in it last season getting out the group stages, so we want to do exactly the same this season.
“We won’t be favourites, it will be a tough match but one we feel we can go and win.
“I’m very excited. You have to look forward to these big games. We’ll get there tonight and train in the stadium and it will feel like a Champions League game.”
Asked if there was anything out of the ordinary about manager Ronny Deila meeting with chief executive Peter Lawwell on Sunday, after being toppled off the top of the table by the Dons, Collins said: “Nothing whatsoever. The manager meets the chief executive every week.
“I wasn’t in the discussion but the talk was about football. That’s what it is always about, that’s why they talk.”