Kamara told his supporters that voting was marred with fraud and the results did not reflect many concerns raised by the party.
President Bio has however called on his rival he defeated in an election run-off to drop a legal challenge against the result.
"I'll very much encourage him to move away from that path and come," Bio told Reuters.
"He is a resource. He has the expertise and the experience. And I think we can put those to work in addition to all that we have to make sure that Sierra Leone is a better place."
While Bio won the presidential poll, his party Sierra Leone's Peoples Party won only 47 of the 132 seats in a parliamentary election. Kamara's APC got 67, a slim majority.
Unconfirmed reports have suggested that Bio would offer Kamara a position in the government in order to placate him
Bio briefly ruled the west African nation as head of a military government in 1996. He replaced outgoing President Ernest Bai Koroma, who could not seek re-election due to term limits.