The 26-year-old is currently on special leave from Rubin Kazan who Celtic have already raided this week for winger Saed Hakšabanović having also signed Carl Starfelt from the Russian side last year.
Ange Postecoglou hinted that a move for Abildgaard was possible after the 4-1 win over Ross County in the Premier Sports Cup on Wednesday night.
"That’s a name, definitely a name,” he said when asked if his was a name he recognised. “There have been a lot of names. There are 24 hours to go, there are still some outgoings we are working on as well. We’ll see how things fall.”
It now appears that the deal is very much on with transfer expert Fabrizio Romano claiming that the move will go through ahead of tonight’s transfer deadline.
He tweeted: “Celtic are now set to sign Danish midfielder Oliver Abildgaard. Deal in place, agreement has been reached now waiting to sign contracts.”
Abildgaard commanded a £1.8million transfer fee when he moved from Aalborg BK to Rubin Kazan in 2019 and he has been capped once by Denmark.
Abildgaard’s signing would take the number of new arrivals at Parkhead this summer to six following on from Mortiz Jenz, Benjamin Siegrist, Alexandro Bernabei, Aaron Mooy and Haksabonovic. In addition, the Hoops also spent in excess of £12million turning the loan deals of Jota and Cameron Carter-Vickers into permanent transfers.