The Sun says Juve are ready to offer £30m for the Republic of Ireland international.
Juve are selling Wojciech Szczesny and have brought in Michele di Gregorio from Monza, but could still be in the market for another goalkeeper.
Kelleher has also been linked with Celtic.
The Irishman is open to hearing from Juve, with his status at Liverpool still as second-choice behind Alisson.
