The Italian guided the Blues to a third-placed Premier League finish and secured the first trophy of his managerial career in the form of the Europa League.
Despite a strong first year in England, Sarri's preferred style of play and stubborn selection policy has drawn criticism in certain quarters, with exit talk swirling since Christmas.
The Chelsea head coach is expected to succeed Massimiliano Allegri at Juventus this summer, having already confessed that he misses life in Italy.
But Poyet, who played at the Bridge between 1997 and 2001, is hoping that Sarri will stay in west London, despite fears he may have already "had enough" of English football.
“Sarri, for me, has done very well,” Poyet told talkSport.
“I know we talked earlier in the season about the style of football, about Sarriball, blah, blah, blah.
“But at the end of the day, he finished third behind the two best teams for many, many years in the Premier League, he won the Europa League and was in the final of the League Cup, which they lost on penalties. So it wasn’t that bad.
“I’m not sure whether it’s Chelsea that want him to leave, or he himself wants to leave.
“It looks to me that he’s had enough of the criticism for the job he’s done and actually he is the one who wants to leave.
“Me? I want Sarri to stay. Having the results they’ve had, I think it’s a good option to have him for another year. But I’ve got a feeling that it’s him who has had enough. I hope not, though.”
Chelsea legend and current Derby County boss Frank Lampard has been touted for a return to his old club if Sarri leaves, after an impressive debut season at Pride Park.
Poyet has urged a former team-mate to seize the opportunity with both hands if he is offered the job, adding: "I think it’s a beautiful job and, if he has the chance, Frank has to jump at it.
“It’s a rare opportunity and it’s a good option. He knows the club, he knows the politics inside the club and he knows you need to have a different way of playing.”