The 22-year-old Serbian international has previously held talks with Cardiff City about re-joining the Welsh club on loan, after first joining the Bluebirds back in January. He made 14 appearances as they secured promotion back to the Premier League, but it appears Palace are looking to hijack any future deal.
The Reds are thought to be open to the idea of Grujic returning to the Cardiff City Stadium for a second spell, with the player keen to establish himself at first-team level.
However, Palace boss Roy Hodgson is not said to be discouraged over his club's chances of securing a late deal for the player. Grujic's international teammate Luka Milivojevic is a prominent figure at Palace, and could make it easier for the player to settle at Selhurst Park.
Grujic has made eight substitute appearances for Liverpool in the Premier League since joining from Red Star Belgrade in a £5m deal in 2016, but has a host of talent ahead of him in the Anfield pecking order.
Crystal Palace could actually turn out to be a great choice for Marko Grujic with Luka Milivojevic being there to help him settle in. Reckon it can be a huge 'selling point' for his agent as well, to get his boy closer to Klopp and surround him with a positive influence.— Miloš Marković (@MilosMarkovic) July 25, 2018
Hodgson is not limiting his options to Grujic, however, and is also said to be looking at a possible move for Chelsea outcast Danny Drinkwater.
The 28-year-old England international made just 12 Premier League appearances for Chelsea last season after struggling with a string of injuries, making a further 10 appearances in other competitions.
German midfielder Max Meyer, who is a free agent after running down his contract with Schalke 04, is also believed to be a target for Palace, but it seems Grujic may be the Eagles most viable option.