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The 33-year-old is surplus to requirements at the Camp Nou and has been told to find a new club.
However, ESPN report the Uruguayan still has a year remaining on his deal and he wants the Catalan giants to pay the remainder should he leave this summer.
Barcelona have also told sources they are willing to let Suarez sit in the stands until his contract expires next summer.
A move to Juventus could be on the cards as new boss Andrea Pirlo looks for a backup to Cristiano Ronaldo.
Terms have reportedly been agreed but a number of obstacles are still preventing a transfer.
Many had assumed Suarez's departure was down to new manager Ronald Koeman but the Dutchman has insisted it was the club's call.
Suarez is Barcelona's third all-time top goalscorer and has been one of their key players since joining in 2014.
Ronald Koeman to Luis Suárez in their telephone conversation a few weeks ago: "For me you could stay but it's the club that doesn't want you." [as]
- barcacentre (@barcacentre) September 11, 2020
His friendship with Lionel Messi has seen them flourish on the pitch and they formed two thirds of the lethal 'MSN' partnership with Neymar until 2017.
Last season saw his influence diminish and he scored 16 goals in 28 La Liga appearances.
He has admitted he would take a reduced role with the Blaugrana - but wants clarity on his future.
"There has been talk of some names that the president suggested, changes that could be made, but nobody has told me that they want to get rid of me," Suarez told AS.
"If that is what the club wants it would be nice if the director responsible for these decisions spoke directly with me. It's better than leaking if I'm one of the players they want to leave. We'll see what happens.
"I want what's best for the club as well and right now my intention is to stay on but if the club decide I'm no longer wanted, I have no problem talking to those that decide."
Do you think Luis Suarez will leave Barcelona this summer?
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Featured Image Credit: PA Images
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