Madrid captain Ramos called a press conference at Madrid's training ground in Valdebebas to "end as soon as possible any rumours that have arisen".
He continued: "I want to make my future clear, I'm a Madridista (Real Madrid fan), I want to retire here and what better way than to rebuild this team?
"The day I leave I'd like to say goodbye through the front door, by winning."
Ramos and Perez had a heated argument in March following Madrid's defeat by Ajax in the Champions League, and their relationship has been strained ever since.
Perez claimed in an interview with Spanish radio station Onda Cero on Monday that Ramos said he had an offer to move to China but the club were unable to afford the transfer. Perez said he replied Madrid would not allow their captain to leave for free.
"I have never wanted to go to China," said Ramos, who added he and Perez held a meeting for "reconciliation" on Wednesday.
"I want to stay in the place I feel loved. Yesterday we fixed everything and we want to build the future of Madrid together."
He added: "The president and I have a father-son relationship, we love each other very much. I do not want to leave, my dream is to retire here, in fact I would be willing to play for free here.
"Who has not argued with their father? But the best solution is to sit face-to-face talking, with nobody else interfering. It's been a long time since I spoke about my contract or money with Florentino."
Real Madrid are expected to complete a deal this summer for Eden Hazard, who said "I think it's goodbye" after Chelsea beat Arsenal to win the Europa League on Wednesday.
"Hazard seems to me a world-class player," Ramos said. "It would be very good for Real Madrid. I don't know, but I hope he can come."