The clubs reached an agreement over Doni's transfer last month but the keeper rejected Liverpool's first offer, believing it to be too low.
However, the player could still be on his way to the Premier League before the weekend.
"In the next 36 hours we will know something definite," Doni's agent Ovidio Colucci said.
"Liverpool are interested in Doni and there is the agreement between the two clubs and with the player.
"Doni is in Italy right now.
"We just have to ratify certain details with Roma.
"If the deal is completed, Doni will join Liverpool on a free transfer and will sign a two-year contract.
"Doni is very happy with the chance to go to Liverpool and play in the Premier League."
The Brazilian joined Roma in 2005 from Juventude in his homeland.
Doni, who is under contract with the Serie A outfit until June 2013, has been an understudy to Julio Sergio in recent seasons, and made 16 league appearances for Roma last term.