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.
Cucurella, 24, however has made a decisive attempt to force through the deal by handing in a transfer request.
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 Skysports.
"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."