It may be too late for the Nigerian defender to be considered for Wednesday's opening Champions League group game against Benfica.
However, Hoops boss Neil Lennon will be relieved to finally welcome the man signed from Israeli side FC Ashdod for £1.2million as the transfer window closed.
Red tape and holidays have prevented Ambrose from returning to Glasgow after he flew back to Israel within hours of putting pen to paper on his contract at Parkhead.
The paperwork necessary to complete his transfer has had the 23-year-old flying between Isreal, Nigeria and Turkey.
But, Lennon is confident the saga is about to be concluded.
The Parkhead boss said: "Efe has had to go to Istanbul to pick up the paperwork he needs, and I am hoping he will be here on Monday."
However, Lennon will have wait a little longer than that because Ambrose's agent, Duda Dahan, said a public holiday in Israel is delaying the final piece of the complicated transfer jigsaw being put in place by one more day.
Dahan said: "Efe will be in Glasgow tomorrow night.
"He has got the paperwork from Istanbul. But, because it is the New Year holiday in Israel, he has to wait until the offices open to get the stamp he needs on his passport.
"This will not be a problem.
"Everything has taken until now to complete because the transfer was only agreed at the end of the window and, since then, Efe has been on international duty."
As a player from outwith the European Union, he requires a work permit.
But, Dahan said Celtic have dealt with this. He said: "The work permit is fine. All we need is this stamp for his passport."
Lennon would prefer not to pitch the new arrival straight into his side without so much as a training session.
Despite the poor performance by Celtic at the weekend when they lost 2-1 to St Johnstone, the manager is expected to consider Ambrose for the bench at best on Wednesday.
Dahan said: "That decision is for the manager to make.
"Efe is fit. He has been playing for Nigeria and has been training in Israel."
The same can't be said of Beram Kayal.
Lennon said the midfielder's ankle injury will again rule him out of Wednesday's game. "Joe Ledley is a doubt, but Adam Matthews will be okay," he said.