The Ghanaian midfielder has just ended a loan spell in Norway with Stromsgodset, with the English champions keen to farm him out again to a bigger club, and that has alerted Celtic to the rising star.
But they will have to battle it out with Belgian champions Anderlecht, who they are also trying to fight off the Hoops to land Markus Henriksen.
And 20-year-old Abu would welcome a move to Scotland. He said: "I am sad to leave Stromsgodset, but I am leaving with a good feeling because we are top of the league and playing great football.
"I am now returning to Manchester City and they want me to play in a stronger league and speed up my development.
"There have been Dutch teams watching me in the stands and Spanish and Scottish teams are also interested."
Abu, who joined City in 2010, had two loan spells with Stromgodset and made a big impression.
He was an important part of the Ghana squad for the African Nations Cup earlier this year and has two years remaining on his contract with the Eastlands club.
Now the Scottish champions are interested in offering him a taste of Champions League football.
As the Parkhead side get set to swoop for Abu, former goalkeeping target Craig Gordon has started training with old club Hearts.
The 29-year-old turned down a three-year-deal with Neil Lennon's men, who have since signed Fraser Forster on a permanent contract.
And Gordon is getting back to full fitness at the Tynecastle club before looking for a new club in England or abroad.
Elsewhere, Kilmarnock target Farid El Alagui has rejected a move to the SPL in favour of signing on for League One side Brentford.
The Moroccan made a big impression at Falkirk in the Irn-Bru First Division last term after signing as a free transfer.
The 27-year-old scored a total of 18 goals for Steven Pressley's side, including a brace against Rangers in a 3-2 victory in the Scottish Communities League Cup.
The Rugby Park side and St Johnstone were both keen to give the former Bergerac forward a chance to play in the top flight, but El Alagui settled on a switch to England's third tier and Brentford gaffer Uwe Rosler said he was happy to secure his signature.
"He has a good goalscoring record and has shown consistency over the past two years," he said.
"I see him as a late developer, he is 27 now and there is more to come from him.
"We expect goals from him – he can add something to our squad."
