New manager Ronny Deila suffered a disastrous third game in charge last week when his side were thrashed 4-1 in a Champions League qualifier against Legia Warsaw, who missed two penalties.
Deila has so far only brought in free agent goalkeeper Craig Gordon and out-of-favour Cardiff forward Jo Inge Berget on loan, with forwards Georgios Samaras and Tony Watt departed and both Scott Brown and James Forrest injured.
Speculation is mounting that goalkeeper Fraser Forster and centre-back Virgil van Dijk will depart, especially if Celtic fail to turn round their three-goal deficit at Murrayfield on Wednesday.
Celtic struggled in the Champions League group stages last season after the summer departures of striker Gary Hooper and midfielder Victor Wanyama, and McAvennie believes the club are in danger of losing disillusioned fans if they do not invest in the team.
Radio Clyde pundit McAvennie, who was helping launch Bauer Media Group's enhanced Scottish football coverage, said: "I'm worried about the season if they don't get through on Wednesday. The fans are only going to take so much if the club is not investing.
"If they don't win on Wednesday I think they will lose van Dijk and Forster, and from what I hear there's a few more want to leave.
"Celtic have got to invest. I don't know who Ronny Deila is.
"I'm worried about Celtic because this is not the Celtic I know.
"I know a lot of people who have boxes and season tickets at Celtic and they are not taking them out this year. That was (because of) last year's team. Nobody is going to watch them if some of those players leave.
"If Gordon is fit, he's a top-class keeper so if they get good money for Forster, they should take it.
"But I don't think they have anyone to take van Dijk's place. There's obviously no-one in the reserves or youngsters because they got battered by Tottenham."
Alarm bells started ringing for McAvennie when Neil Lennon quit after more than four years as manager following the end of last season.
"I had a feeling he was going to go," the former West Ham player said. "I thought he was going to go to Newcastle because Alan Pardew was under pressure. But Newcastle went and gave him investment and he has bought players and they might have a good season now.
"That's what happens when you back the manager. Celtic never backed Lenny."
Berget went straight into the team in Poland two days after signing while last season's main strikers, Leigh Griffiths and Anthony Stokes, started on the bench. And McAvennie feels that was a strange decision on two counts.
"One good striker makes all the difference, but they brought in another boy who couldn't get a game for Cardiff and he made the team," he said. "It's not good for morale if they are putting him right in the team."
However, he still feels Celtic have a chance of turning round their deficit.
"If you can get the ball to Kris Commons and get Griffiths and Stokes playing, then they have a chance. They scored four, it could have been six or seven, but I'm sure Celtic have a better team than them so there's no reason why they shouldn't beat them."