The Reds were linked earlier this month with the 22-year-old, but Sky Italia claims Genoa have put a £10million price tag on his head.
Perin came up through the academy system at Genoa but he had season-long spells on loan at Padova and Pescara.
He has represented Italy at every level from the under-17s right up senior level. He made his full international debut in the friendly win over Albania last month.
Perin is seen as a replacement for Simon Mignolet, who has been dropped by manager Brendan Rodgers. Liverpool have also been linked with Fiorentina’s Brazilian keeper Neto.