The Daily Star says his position has been under threat for months.
But Chelsea's top choice to replace him Luis Enrique doesn't want to join them until the end of the season.
The champions dropped out of the top four on Sunday following a 2-1 defeat at Manchester United.
Conte and owner Roman Abramovich and his Chelsea board have not seen eye-to-eye throughout this season, with the Italian angry at having little say in transfers.
But it appears that Abramovich has now accepted that Conte is the best option to see out the season.