Boss Neil Lennon has confirmed that the Scottish champions have nearly agreed a deal for Mbombo as he prepares for life after Tierney.
The Scotland international has been the subject of interest from Arsenal, who tabled an initial £15m bid for the left-back.
This was way short of the value that Celtic have placed on the defender, but the Gunners are still expected to come back with a revised offer.
According to the Daily Express, however, Serie A side Napoli have the edge over Arsenal as they try to beat Unai Emery to Tierney’s signature.
“We are not too far away and he will be another good addition to the squad”
Meanwhile, Mbombo is jetting in to Glasgow regardless as Lennon looks to tie down the 22-year-old before Thursday’s Champions League qualifiers deadline.
He said: “We are not too far away and he will be another good addition to the squad.
“He gives us competition at left-back, he’s strong, athletic and good on the ball. He fits the criteria and hopefully he’ll be in before Thursday.”
Lennon insists Celtic want closer to £25m if they are to let Tierney leave Parkhead this summer.
But he admitted that the uncertainty surrounding the 22-year-old could affect the defender whether he leaves or not.
Lennon said: “The first offer wasn’t what we’d look for and I don’t think we’ve had another one yet. We have a price. It can be unsettling but he’s been okay.”
Tierney is one of the most highly-rated talents north of the border, having won the three consecutive SPFA Young Player of the Year awards prior to last season.
He is currently undergoing rehabilitation from a double hernia operation he had in May, having only managed to play a handful of matches since the start of December.
But there is a belief that he may have already played his last game for his boyhood club.
Lennon also revealed Zenit St Petersburg have yet to make their expected £9m move for Tom Rogic – but he admitted he wants to keep the Aussie ace.
He said: “I haven’t heard anything and there’s been no enquiries. I want to keep all of my good players.”