The Spain left-back, 24, handed in a transfer request last week.
The Seagulls had already rejected one bid from City and see £50m as the benchmark – the fee they sold defender Ben White to Arsenal for last summer.
Sources say it would take an exceptional offer to consider a sale as they see Cucurella as a better player.
Pep Guardiola’s side are now exploring other options as the fee is deemed too high.
Cucurella had been identified by City as a replacement for Oleksandr Zinchenko, who left the club to join Arsenal this summer.
Cucurella has a contract until 2026, having only joined Brighton in August 2021.
He has made 35 Premier League appearances for the club, scoring one goal, since joining from Getafe last year in a £15.4m deal.