The goalkeeper has a deal to 2027 with Roma, but is among the lowest paid players in the squad.
The father of Svilar is now in talks with Roma chief exec Lina Souloukou, who recognises the player deserves a pay-rise.
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The keeper is set to be backed as Roma's No1 next season, with Rui Patricio to be moved on.
TMW says Roma are eyeing Atalanta goalkeeper Pierluigi Gollini, who is currently on-loan with Napoli, as a replacement for Patricio.