The player has arrived in Glasgow for talks and to undergo a medical as the Scottish Premiership club try to secure a work permit for the 19-year-old.
Eboue joined the Russian Premier League club in 2014 from Armenian outfit Shirak.
He has made 18 appearances this season for the side sitting fifth in Russia's top flight.
Now Eboue could become Celtic's first signing during the January transfer window.
He was recently handed his first Ivory Coast call-up but has yet to make his international debut.
Speaking last month, Celtic manager Brendan Rodgers said he was keen to bolster his squad in January.
"We have a month [of the transfer window], and the first three weeks we'll be resting for a little bit and working very hard in some warm weather," Rodgers said in reference to the winter break.
"I expect us to do some business, [whether it's] the beginning or the end, as long as it's done that's all I'm really concerned about.
"There will be [players leaving], that's a natural way when players aren't playing so much, especially if they're mid-20s and beyond. They want to play regularly. Some will come in to help the squad that's already here.
"I can't have any complaints about any player, the attitude. We've got a number who are outside the squad, but their professionalism and devotion to what we're doing here - they're at the best weight they've ever been - is a huge testament to them."
Celtic have agreed a fee in the region of £3m with Krasnodar for Ivorian midfielder Kouassi Eboue.