The 31-year-old, who has been with the club since 2005, has one year left on his contract but has been widely tipped to leave this summer.
"I spoke to [Roma] yesterday and they confirmed they are holding talks with Liverpool to reach an agreement regarding the transfer of Doni," said Doni's agent Ovidio Colucci."
"Once an agreement is reached between the two clubs, we will begin negotiating personal terms with Liverpool. Doni would be very happy to join Liverpool and play in the Premier League."
Liverpool are also courting Sunderland's Josh Henderson and Blackburn's Phil Jones.