The Blues hierarchy had pinned their hopes on landing Zinedine Zidane.
Owner Roman Abramovich has no desire to sack Sarri before the end of the season.
But there is a growing feeling the Italian is not a natural fit for the club.
Fans revolted against Sarri during last month's FA Cup defeat by Manchester United and are now prepared to boycott home games.
Sarri will only go before the end of the season if Chelsea crash out of the Europa League and suffer a collapse in their battle for a top-four place.