The rangy midfielder spent last season on loan at the KC stadium from Spurs, where he played a key role in the Tigers’ surge to safety and shock run to the FA Cup final.
There has been talk that Steve Bruce may look to keep the 24-year-old, yet a move to nab his signature is stalling due to a disagreement over a fee.
This has opened the door for Palace, with manager Tony Pulis keen to snap up the Spurs star.
The once Stoke chief is set to be backed in the transfer market after his miraculous 2013/14 at the helm of the Eagles, and wants to add more quality to his central ranks.
Livermore has previously spoken of his willingness to stay at Hull, but remaining in London may be an option that appeals to him.
Spurs are eager to ship out their academy graduate, with Mauricio Pochettino already possessing a wealth of midfielders.
Michael Dawson is also a target for Palace.