NFL teams had until 8am Tuesday AEDT to submit a claim for former Australian rugby league star Hayne with speculation the Dallas Cowboys, Seattle Seahawks and Detroit Lions were interested.
The 49ers were ravaged by injuries in Sunday's 27-6 loss to the St Louis Rams, with Bush, who took over punt return duties from Hayne, and Davis, a rookie running back who some believe should have been fired instead of Hayne, unable to finish the game.
Bush suffered a suspected season-ending torn anterior cruciate ligament, Davis injured his hand while the 49ers' best running back, Carlos Hyde, was ruled out last week with a stress fracture in his foot.
The 49ers' playoff hopes appear dead after sinking to their sixth defeat in eight games and languishing at the bottom of the NFC West division.