Brendan Rodgers is to make the 31-year-old his fifth signing of the summer.
“We were always on the lookout for a really aggressive number eight to play midfield,” Rodgers said ahead of Celtic’s match with Suduva.
“Youssouf was exceptional last year, he was very keen to come and play at a huge club and fits the profile and has the experience of being up here.
“All being well, it’ll be a really good move for us.”
The Congolese midfielder impressed during his time at Rugby Park last season where he netted goals against Celtic and Rangers and made 20 appearances to help Killie finish in the top six.
Mulumbu is currently a free agent but has interested a number of clubs including English Championship side Swansea City and Ligue 1 outfit Bordeaux.
He is expected to be the first of a few signings made by Celtic before the close of the transfer window.
The future of Moussa Dembele is still to be resolved with Lyon keen on bringing the player to France.