The person spoke on condition of anonymity because the agreement isn't expected to be announced until Monday.
Kubiak replaces John Fox, who was ousted last Monday following Denver's loss to Indianapolis in the playoffs.
Elway and team president Joe Ellis met with Kubiak in Houston for several hours Sunday.
Kubiak was Elway's backup QB from 1983-91 and Denver's offensive coordinator from 1995-2005 under Mike Shanahan.
Kubiak, who was the Ravens' offensive coordinator last season, went 63-66, counting the playoffs, as head coach of the Texans from 2006-13.