The Serie A side accepted a bid from the Reds last month, only for the Brazilian to turn down their contract offer, deeming the wages to be too low.
Following Roma's sale of Doni's first-choice rival Artur to Benfica, the 31-year-old is by no means pushing for a move to England.
However, his representative Ovidio Colucci has revealed Liverpool's interest persist, and suggests a decision on his client's future could become clearer next week.
"At the moment there's no news on the situation of Doni," he told Calciomercato.it.
"The hypothesis of Liverpool is still possible. Towards the middle of next week we will know more, since I scheduled a meeting with the leadership of Roma."
Should Doni remain at Roma, he faces competition from Bogdan Lobont and compatriot Julio Sergio for the number one spot.