The Parkhead outfit have had talks with the former Bournemouth boss about possibly replacing Neil Lennon, who resigned as Hoops manager in February with Rangers on the way to preventing their Old Firm rivals winning a 10th successive Premiership title.
Reports subsequently claimed Howe is waiting until the summer before he makes a decision about his future.
However, the 61-year-old former Scotland, Celtic and West Ham striker, speaking at Hampden Park to promote coverage of the Rangers versus Celtic Scottish Cup tie exclusively live on Premier Sports this weekend, said: "He could say to Celtic, I am coming or sign paperwork to say I'm coming.
"If it is true that he is going to speak to Crystal Palace and Newcastle then I would say no.
"I don't know if I want him because his heart is not in coming up here then.
"I played in England a long time and people don't realise down south how big a club Celtic are, and Rangers for that matter.
"They are huge clubs and they won't know until they get up here and walk through the door and realise how big they are.
"He is a good manager and of course people will be interested in him.
"But I just hope Celtic have not put all their eggs in one basket and relied on Eddie Howe because there are a lot of good managers out there.
"You can't wait until June for Eddie Howe to say no. Celtic is too big a club for that, they shouldn't wait for anyone in that respect.
"I am sure Dermot (Desmond, principal shareholder) and Peter (Lawwell, outgoing chief executive) and the new chief executive coming in (Dominic McKay) will be talking to different people, I am sure of it.
"Celtic is a massive club so they won't be just talking to one guy, surely not?"
McAvennie believes Scottish Cup holders Celtic are in fine fettle for the trip to Ibrox following their convincing 6-0 Premiership win over Livingston at Parkhead on Saturday, their biggest league victory of the season.
He said: "That was a strong result for Celtic at the weekend. Livi are not an easy team to beat.
"The players are all playing for each other now, you seen that at the weekend.
"Maybe they are playing for a move or to stay for next season, who knows and who cares, as long as they turn it on when it matters against Rangers."
:Frank McAvennie was speaking to promote coverage of Rangers v Celtic exclusively live on Premier Sports this weekend. Premier Sports is available on Sky, Virgin TV and the Premier Player. Prices start from £10.99 per month.