The record seven-time Europa League winners are languishing in 12th in La Liga, although when Sanchez Flores was appointed in December they were battling relegation in 16th.
Sanchez Flores' contract was set to expire in June 2025 but Sevilla chief Jose Maria del Nido Carrasco told reporters both parties agreed the coach will depart.
"I want to thank him, the club was in a very difficult situation and he was capable of turning it around with a lot of hours of work," the president told a news conference Saturday.
"He achieved an objective that, although it hurts us to say it, was difficult."
The president also said he would offer captain Navas a contract for life, and the Spain international quickly confirmed he would stay with the club he "is in love with".
On Thursday Sevilla announced their record appearance-maker would depart at the end of the season, before Navas released a statement on social media late Friday criticising Sevilla.
The 38-year-old said he had "received no call" from the club about renewing his deal which expires in June.
Navas denied having agreed any deal to play elsewhere and said he wanted to focus his "body and soul" on playing for Spain at Euro 2024 if selected.
"There must have been a misunderstanding for which I will take responsibility, because I know Jesus is suffering a lot," said the Sevilla president.
"If I sit down with Jesus for a minute, we will turn the situation around.
"From here, I offer him a lifetime contract so he can play here as long as he likes, and later work here in whatever he wants, because he deserves to be at Sevilla as long as he wants."
Swiftly Navas issued a statement saying he had agreed to play on in the south of Spain for a few more months.
"Not for a second did I doubt saying yes to the proposal offered by the president of my Sevilla," said Navas.
"Yes, I want to retire (here)... I willingly accept staying linked to the club that I am in love with."
Navas said he wanted to keep playing until December, before retiring, with his salary until then to be donated to a charity.
