The 25-year-old international is a free agent following the expiry of his contract at AS Kigali. He is also said to be in talks with top Ethiopia club Coffee.
"He has an offer to extend his stay in Rwandan but will listen to offers from top Kenyan clubs," a source told Goal.
An established defender, who can also put in a shift at right back, Kayumba, who has represented his country 24 times, captained AS Kigali to the Peace Cup trophy in 2016.
His club finished in the third position during the just concluded Rwandan league season.
Kayumba's impending arrival could signal the sudden interest in the Kenyan league by its Rwandan equivalent. Gor Mahia trio Karim Niziyigimana, Jacques Tuyisenge and Francis Mustafa all recently moved to Kenya from Rwanda.
Kiyovu SC coach Casa Mbungo is another Rwandan, who has been linked with a move to AFC Leopards.