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Sky Sports News understands the development has not changed Brighton's stance on holding firm in their valuation for the 24-year-old.
The Seagulls have rejected a £30m bid for the full-back from defending Premier League champions City and value the Spaniard at £50m.
Cucurella enjoyed a hugely successful first Premier League season after joining from Getafe in a deal worth around £15m last summer and was named Brighton's player of the season.
Brighton do not want to let Cucurella leave and it would take a huge offer for them to sell as he has four years remaining on his contract at the Amex. However, the left-back has now pushed for a move by asking to be sold.
Manchester City manager Pep Guardiola announced interest in signing a left-back but refused to be drawn into the latest update on Cucurella's future.
"If possible, yes [we want to sign a left-back]," said Guardiola in a press conference before City's Community Shield game against Liverpool on Saturday.
"But if not, we stay. Joao [Cancelo] can play there. Josh [Wilson-Esbrand] and Nathan [Ake, too]. I never complain about the squad I have since the first season. What the club can do is the benefit for the club not me."
When asked about Cucurella specifically, Guardiola said: "He's a Brighton player, I cannot say anything else."
The City manager also announced that defender Aymeric Laporte will be out until September after undergoing knee surgery.
Potter: Unless clubs agree, there's nothing to say
Brighton manager Graham Potter told Sky Sports News this week that while he respects the notion that players like Cucurella have career ambitions and families to think about, a move for his left-back is not worth talking about until the two clubs can agree on a fee.
"You can talk and respond all you want to media suggestions but until there is something where the clubs and the player agree, there is nothing to say," Potter told Sky Sports News.
"The guys know that it's part of life and the noise around football, certainly at this level. You can control only what you can control. It's part of life and it's as simple as that.
"They're human beings, while the window is open, the possibilities can happen. That's for all the players, you have to respect that they have families, careers, ambitions and things like that.
"At the same time, you know it's part of the game and something they have to deal with. We just have to focus on the next training session and match. That's the only thing they can do."
Carra: Left-back the problem position under Pep
Jamie Carragher says Man City have lacked an "outstanding" left-back during Guardiola's time in charge but that Cucurella would provide healthy competition should he move to the Etihad.
Speaking to Sky Sports News, the former Liverpool defender said: "I think Man City have always - really since Pep's been there - lacked a real outstanding left-back. It was always the position where they've had sort of different players at different times.
"You're never quite sure who plays there every week, where you know it's Kyle Walker playing right-back. [Joao] Cancelo's gone in there and been as good as anyone really but I'm sure Pep would like a left-footer there.
"Zinchenko's moved onto Arsenal so when you ask 'is he worth it', Man City normally spend £40-50m on full-backs. They've done that notoriously since Pep came and like any transfer and what you pay - if you play well you think he's worth it.
"He looks a classy operator when you see him in a Brighton shirt. He plays a similar type of football to Manchester City in possession, so I'm sure he'll be a good signing.
"It's something that Man City possibly need now with Zinchenko moving on and having a bit more competition for Cancelo, who can then provide competition on the other side for Walker."
Man City fixtures: West Ham visit live on Sky Sports
Manchester City kick off their Premier League title defence with a visit to West Ham on Sunday August 7, with the game live on Sky Sports.
The first Manchester derby of the season is at the Etihad on October 1, before City travel to title rivals Liverpool on October 15.
The champions' final game before the season takes a break for the winter World Cup is at home to Brentford on November 12 and when the campaign resumes, they face Leeds at Elland Road on Boxing Day.
City have a tough start to 2023 with tricky-looking back-to-back clashes at Chelsea and Manchester United on January 2 and 14 respectively.
As the campaign draws to a close, Pep Guardiola's side have their return meeting with Liverpool, at the Etihad, on April 1 and then host Chelsea on May 20, before concluding at Brentford on May 28.
Follow the summer transfer window with Sky Sports
Who will be on the move this summer when the transfer window opens on June 10 and closes at 11pm on September 1?
Keep up-to-date with all the latest transfer news and rumours in our dedicated Transfer Centre blog on Sky Sports' digital platforms. You can also catch up with the ins, outs and analysis on Sky Sports News.