The pressure on the 41-year-old has intensified dramatically following a dismal 3-0 loss to Brighton on Tuesday, marking the Blues' fifth consecutive Premier League defeat without finding the net.
The performance on the south coast was particularly dire. Rosenior's side managed only six shots throughout the game and failed to register a single one on target.
Following the match, travelling supporters at the Amex Stadium voiced their frustrations, calling for his removal.
Chelsea face huge cost sacking Rosenior
Should the Chelsea board heed these calls, the decision could come with an eye-watering price tag.
According to reports from The Sun, sacking Rosenior could cost the club as much as £24 million.
This figure is based on his reported £4 million-per-year salary and a long-term contract that runs until 2031, assuming no specific release clauses are in place.
This potential payout follows a recent pattern of costly managerial changes at Stamford Bridge, with Graham Potter, Enzo Maresca, and Mauricio Pochettino all receiving significant compensation packages upon their departures in recent years.
With just four league matches remaining, Chelsea are languishing 10 points adrift of the Champions League qualification spots.
Their next major test is an FA Cup semi-final clash against Leeds United at Wembley on Sunday.