Madrid announced on Wednesday that captain Ramos, 35, will leave when his deal runs out at the end of this month.
The defender broke down in tears during a farewell ceremony on Thursday -- attended by club president Florentino Perez, board members, club staff and Ramos' family -- at the Real Madrid City training complex.
"I never wanted to leave Real Madrid, I always wanted to stay here," Ramos said. "In the last few months the club made me an offer of one year, with a pay cut. I want to highlight that money has never been a problem, the president knows that.
"It wasn't a financial problem. They offered me one year, I wanted two. I wanted calm and continuity for my family ... In the latest conversations, I accepted the offer with a pay cut, and I was told there was no longer an offer. I was told that despite having given the OK to the proposal, it had an expiry date and I didn't realise ... It surprised me."
Ramos refused to go into details of talks with Perez over a new deal, which had dragged on throughout last season without an agreement being reached.
"They're private meetings that I want to keep private out of respect for the club, the president and myself," he said. "But it's true that it was communicated to me just a week ago, via my agent, that the offer had expired without us knowing.
"Nobody had given us an ultimatum ... I thought there was a possibility to accept the last offer they had put on the table ... They're questions I can't answer. The club has its priorities."
Ramos described his relationship with Perez as "extraordinary" and said he was "eternally grateful."
"I'll focus on the affection and the last hug he gave me today," he said. "I'm not someone who holds onto resentment. Misunderstandings can stay in the past, I'll focus on the good things."
"Today isn't an easy day," Perez had said earlier. "You've been someone very special for me and we've suffered and fought together for Real Madrid's recent history. Above all, I want you to be happy wherever you are... We're convinced that a legend like you will always be a great ambassador for Real Madrid."
"I haven't thought about any team at the moment," Ramos said. "It's true that since January, when I was free and on the market for contractual reasons, we've had calls to my brother from interested clubs, but at no time I've thought about leaving. Now we'll look at a good option for me."
He added: "Sevilla is the other club of my heart, but right now I'm not considering that option and Sevilla aren't considering it either. As for Barca, it's a rotund no. It's impossible to see Ramos in a Barca shirt."
The centre-back is widely recognised as one of Real Madrid's most influential players, making 671 appearances and scoring 101 goals since signing from Sevilla at the age of 19 in 2005.
He scored one of the most significant goals in the club's history -- a 93rd-minute equaliser in the 2014 Champions League final against Atletico Madrid -- and became Real captain in 2015 after Iker Casillas' departure.
Ramos has won a total of 22 trophies with Real Madrid, including four Champions Leagues, five La Liga titles and two Copas del Rey. He was also a member of Spain's winning teams at Euro 2008, the 2010 World Cup and Euro 2012.
An injury-hit 2020-21 season -- making 21 Real Madrid appearances in all competitions, just five of them coming in 2021 -- saw him left out of Spain's Euro 2020 squad by coach Luis Enrique.