Sarr is understood to be desperate for a fresh challenge having enjoyed a hugely successful spell at Selhurst Park, helping the Eagles win both the FA Cup and the Conference League.
Those close to the player feel he has run his race at Crystal Palace and that the time is right to move on, having served the club well enough that a transfer generating a sizeable fee would represent a fair outcome for all parties.
The 28-year-old scored 21 goals in all competitions last season, following 12 the year before, establishing himself as one of Palace's key attacking outlets during a period that saw the club lose several important figures, including Michael Olise, Eberechi Eze and, more recently, Marc Guéhi.
Earlier in the summer, Turkish clubs Fenerbahçe and Galatasaray explored moves for the winger, though Sarr was understood to rank behind both Marcus Rashford and Rafael Leão on their respective shortlists.
Leão's subsequent move to Galatasaray has now cleared a path for Sarr to pursue Liverpool instead, a switch he would reportedly welcome.
Sarr still has three years remaining on his Palace contract, leaving the Eagles in a strong negotiating position despite his stated desire to leave. An offer has been made for the player, though sources at Palace have declined to confirm the exact figure involved. Liverpool and Sarr's representatives are now waiting on a response from Selhurst Park, with movement possible before the window shuts.
The pursuit of Sarr comes as Liverpool continue to weigh up their options across the forward line, with manager Andoni Iraola keen to add further competition out wide following the departure of Mohamed Salah.
Any deal for Sarr would add proven Premier League experience to a Liverpool squad that has otherwise leaned heavily on younger recruits this summer, and comes after the club's long-running pursuit of an alternative winger fell away when talks over a rival target stalled.
Sarr's familiarity with English football, having previously played for Watford and Metz before joining Palace, would allow him to slot in with minimal adaptation period, something Liverpool are understood to value given how late in the window any deal would be completed.