City had a £30m bid rejected for the Brighton player of the year earlier in the window and this week threatened to walk away from the deal.
Brighton remain stoic in their valuation and want £50m for the star man who is contracted to the Seagulls until June 2026.
Carragher was impressed by Cucurella last season and believes his style of play is perfect for City.
“Since Pep has been there, Man City have lacked a real outstanding left back,” he said earlier this window to Sky Sports.
“It has always been a position for different players. João Cancelo has done as well as anyone.
“I’m sure Pep would like a left footer in there and Oleksandr Zinchenko has gone to Arsenal. Man City usually spend £40m to £50m on full backs since Pep came.
“Like any transfer, you have to see if he plays well and is he worth it?
“He looks a classy player in the Brighton shirt and they play similar type of football to Manchester City.
“I’m sure he would be a good signing for Man City and a player they possibly need and give competition for Cancelo.”
Check out the latest Premier League transfer speculation below...
1. Villa in Luiz talks
Aston Villa are in talks with Douglas Luiz over a new contract and are hopeful of striking an agreement with the Brazil international. (Daily Telegraph)
2. Pavard willing to join Chelsea
Bayern Munich defender Benjamin Pavard would be willing to join Chelsea if a suitable offer is made. (Bild)
3. Januzaj training alone
Reported West Ham target Adnan Januzaj is training alone as he makes a decision over his next career move following his departure from Real Sociedad. (El Diario Vasco)
4. BURNLEY, ENGLAND - APRIL 21: Maxwel Cornet of Burnley warms up prior to the Premier League match between Burnley and Southampton at Turf Moor on April 21, 2022 in Burnley, England. (Photo by Clive Brunskill/Getty Images)
Newcastle United are ready to trigger Maxwel Cornet’s £17.5m release clause with Burnley. (talkSPORT)