The 31-year-old midfielder left Kilmarnock at the end of last season and is currently without a club after making 18 appearances in his one Scottish Premiership campaign.
Numerous reports which emerged on Thursday claimed that Mulumbu is undergoing a medical in Glasgow ahead of signing for the Parkhead side.
"We're always on the lookout for a really aggressive number eight to play in midfield," Rodgers is quoted by The Sun as saying. "We felt that's what the team needed, we were down a midfield player.
"I've seen Youssouf in the Premier League and obviously the Premiership up here and he was exceptional for Kilmarnock.
"So he's one that was very keen to come, play for a huge club and fits the profile and has experience of being up here. Hopefully all being well it will be a really good move for us."
Mulumbu, a Democratic Republic of Congo international, previously played for Norwich City and West Bromwich Albion, and reportedly held talks with Swansea City earlier in the transfer window.