The Daily Star says Enrique has already agreed a contract with the Blues.
However the the Spaniard will not take charge until the summer.
He's determined to have the whole campaign off, as he takes break following his exit from Barcelona at the end of last season.
That could ensure Conte stays in a job until May, despite Chelsea's nosedive in form.
The Blues have lost 3-0 at home to Bournemouth and 4-1 away at Watford in their last two games to put a severe dent to their Champions League qualification hopes.
Chelsea are still alive in the race for a top four finish though, and Conte looks set to be given the chance to turn things around.