According to a report from Sky Sports Switzerland, influential figures within the Stamford Bridge leadership structure harbour doubts over whether Rosenior will remain in charge beyond the current campaign.
The club, owned by the consortium led by Todd Boehly and Behdad Eghbali under Clearlake Capital, is keen to establish long-term stability after several managerial changes since the takeover.
Those ambitions could clash with the pressure to deliver immediate results. Chelsea currently sit sixth in the Premier League table, two points behind fourth-placed Aston Villa with eight league matches remaining.
While fifth place may still offer a route into the Champions League, the margin for error is slim, and failure to qualify would represent both a sporting setback and a financial disappointment given the club's heavy investment in the squad.
Recent results have done little to ease the tension. Chelsea suffered back-to-back defeats against Paris Saint-Germain and Newcastle United, including a damaging 1-0 home loss to Newcastle on Saturday.
The decisive moment came in the first half when Joe Willock broke through on goal and squared the ball for Anthony Gordon to tap in after Chelsea failed to defend the move effectively.
Rosenior only took charge in January after replacing Enzo Maresca and initially enjoyed a promising start, winning seven of his first nine matches. However, the coming weeks are viewed internally as decisive for the direction of the project.
Real Madrid have received a significant lift ahead of their Champions League round-of-16 second-leg clash with Manchester City on Tuesday, with both Kylian Mbappe and Jude Bellingham included in the squad traveling to England.
Chelsea have been handed the largest fine in Premier League history and a suspended transfer ban after admitting to historical financial rule breaches involving more than 47 million pounds in undisclosed payments during the Roman Abramovich era.
Chelsea captain Reece James could be sidelined for several weeks after suffering another hamstring injury, head coach Liam Rosenior has confirmed.
Eddie Howe praised Newcastle's "defensive mentality" after his side secured a hard-fought 1-0 victory over Chelsea at Stamford Bridge on Saturday.
