Pellegrini has a little over a year left on his contract and so far he has refused to sign a new deal with the Serie A giants to extend his stay at the Stadio Olimpico.
He is a key part of the Roma squad and the club are keen to hold on to him beyond the approaching summer transfer window.
Liverpool are believed to be interested in taking him to Merseyside as a potential replacement for Georginio Wijnaldum, who could move on at the end of his contract.
But the player has not made a decision and, according to Italian outlet Calciomercato.com, he has pushed all talks over his future back to the end of the season.
Pellegrini wants to speak to the Roma hierarchy and understand the ambition of the club and his role in the project under Jose Mourinho.
The Italian giants want to tie him down to a new deal, but will have to convince the player about their future ambitions.
Liverpool are likely to keep an eye on developments in the eternal city regarding Pellegrini and his future at Roma.