The country’s two biggest sides Al Ahly and Zamalek are both fighting for the Ivorian’s signature.
Press Association Sport understands a deal worth an initial £750,000 has been agreed with representatives of both clubs - with the fee rising to near seven figures if add-on clauses are triggered.
Coulibaly will now have the final say on which club he joins.
The Rugby Park club rejected Zamalek’s £500,000 bid earlier this month but the sale would represent a remarkable piece of business for a player released by Peterborough just last summer.
He has 11 goals to his name this season, with an audacious 40-yard strike at Celtic Park and stunning over-head kick against Dundee already tipped as goal of the season contenders.
But his loss would be a major blow for Lee Clark and his team’s survival bid.
Tenth-placed Killie have scored just 16 Ladbrokes Premiership goals this term - with eight of them coming from Coulibaly.