As per the Press and Journal, the Bhoys are keen on signing Mulumbu this summer, when his West Brom contract expires.
The 28-year-old midfielder has made 211 appearances for the Midlands club in the six years he has spent at the Hawthornns, but his contract runs out at the end of this season, and it not set to be renewed.
Manager Tony Pulis is looking to revamp his playing squad ahead of next campaign, with his first pre-season planned at the club, and he does not see the DR Congo international as having a future at the club.
This has opened the potential for a move to Celtic, who have now shown an interest in signing the midfielder this summer.
Ronny Deila is preparing to build a squad who can compete on four fronts, having struggled with the game load this season. Mulumbu is an experienced and talented midfielder, well regarded for his fitness, who would be an excellent signing.
Being 28-years-old, it is likely that he would look for a transfer which could guarantee consistent playing time - something that is not guaranteed at Parkhead.
However, the draw of Champions League football, and almost guaranteed silverware could see Mulumbu desire to move north of the border, and the Bhoys could do a lot worse that sign a player who has proven himself in the Premier League on many occasions.