Loftus-Cheek focused on himself at Chelsea
The 22-year-old England international is keen to stay at Stamford Bridge this term after returning from an encouraging loan spell with Crystal Palace.
New Blues boss Sarri, speaking after Tuesday's friendly win over Lyon, praised Loftus-Cheek as "potentially a great player" in a sign he is set for a first-team opportunity.
However, the homegrown youth product - who has outlasted Jose Mourinho, Guus Hiddink and Antonio Conte since his 2014 debut - is concerned only by meeting his own expectations.
"At the moment I am at Chelsea, my mind is on playing well here and improving here," Loftus-Cheek told reporters.
"I will play well for myself and improve myself. Whoever the manager is, it doesn't change how I want to play.
"I will always do my best and that's just my mindset to improve myself - however that comes across to the manager is what it is. I look to play well for myself first.
"There is still a lot for me to work on, there's a lot of room for improvement so I will carry on working and hopefully I can do well this season."
Ruben in action at the Bridge tonight. #CFCvLyon pic.twitter.com/OCwfEbHxMh
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Loftus-Cheek's post-World Cup exploits appear to be winning over Sarri, who believes the midfielder can have a long career at Chelsea.
"He has to improve from the tactical point of view," Sarri said.
"But I think that's not a problem. He's very young. I think he can be a very beautiful player for us."