The Daily Mail says Fortune is expected to make a final decision on his future before the end of the week after the Parkhead club held further talks with his French employers to try to secure a deal reportedly worth around £3.8million.
The 27-year-old is also considering the personal terms on offer from various suitors, with Celtic and Hull City now the firm front-runners in the chase for his signature after Portsmouth, Wolves and Fulham were also linked with the former Utrecht forward.
Fortune has been aware of Hull's interest for more than a fortnight, but has stalled on accepting their offer until he knows what is on the table from the Parkhead club.
He has never hidden his desire to work again with Tony Mowbray after a loan spell at West Bromwich Albion last season, but is also keen to land the best financial deal. "The clubs are talking and we are discussing everything," said Fortune's agent, Michael Manuello, yesterday.
"No deal has been struck yet but we're close. I'd expect a decision by the end of the week. Marc doesn't have to meet Tony Mowbray to know what it will be like to play for him at Celtic because he already knows the manager very well and trusts him.
"But he does have a choice to make. He has to weigh up all his options."