The Mirror says Puel is already on the brink at Leicester — with Roberto Martinez and Henry in the frame to replace him.
The Foxes lost 2-1 to Manchester United in their opening game of the season on Friday.
The concern in the Leicester boardroom stems from the team's finish to last season when they lost five from their last seven games with just one win – and that has left the Frenchman clinging to his job.
Belgium boss Martinez, who was at Friday's game at Old Trafford, will be considered for the job if Puel fails to turn things around and his assistant, Henry, could link up with him at the East Midlands club.
However, Henry has made no secret of the fact he wants to become a manager and the Leicester appeals to him as well.