Mack's agents confirmed the move to ESPN.
The Broncos (2-5) play the Jacksonville Jaguars (2-5) on Sunday at Wembley Stadium.
The signing comes in the aftermath of the injury to Broncos RB Mike Boone, who left the Week 7 loss to the New York Jets and was in a walking boot postgame.
Denver contacted Mack's agents -- and the 49ers -- before the game against the Jets had even ended, per ESPN.
Mack appeared in two games for the 49ers this season, playing just five total snaps on special teams.
The Broncos have already lost Javonte Williams (ACL) for the season and new RB1 Melvin Gordon has been largely ineffective.