The 22-year-old is something of an unknown quantity, having played for the French under-20 national side 11 times, and just 20 times for Genk since joining from Metz last summer.
Koulibaly is also capable of playing as a defensive midfielder, although Moyes is keen to strengthen his defence first and foremost.
United have endured a tough start to the season which has seen them take just seven points from a possible 18.
Moyes has watched his side lose three of their last four Premier League games, conceding seven goals in the process.
The former Everton boss endured a difficult summer transfer window in which he was only able to make one major signing, so has begun identifying January targets as early as possible in order to avoid the same mistakes.
However reports suggest noisy neighbours Manchester City are set to rival United for the signing of Koulibaly, as well as 18-year-old St Etienne defender Kurt Zouma.
Manuel Pellegrini has made signing a new centre half his top priority, having watched his side ship three goals on three seperate occasions this season – all of which have ended in defeat.
The powerful 18-year-old was watched in action by City’s top scout Gary Wirthington, with the club expecting £7million to be enough to secure his move to the Ethihad in January.