The Blues want England star Hart to replace Thibaut Courtois – if he joins Real Madrid next summer.
But the Express say Liverpool boss Jurgen Klopp wants Hart too.
He is rated at around £10m after being frozen out by Manchester City – and his £120,000-a-week wages won’t cause a problem to either interested club.
Tottenham were also keen in case Hugo Lloris decided to leave – but he’s now signed a new long-term contract.
Hart is currently on a season-long loan to Italian outfit Torino and feels let down by Pep Guardiola.
Guardiola froze him out after arriving at the Etihad and instead swooped for Claudio Bravo.
City are willing to let Hart leave the club but want him to move abroad.
Etihad chiefs don’t want to strengthen any club that they see as direct rivals.
Courtois, 24, spent three seasons at Real’s fierce cross city rivals Atletico before returning to Stamford Bridge and ousting Petr Cech.
But Courtois knows that with Keylor Navas the current first choice, he has the opportunity to establish himself as the long-term number one at Real.
Klopp, meanwhile, has major doubts over the quality of both his keepers – Simon Mignolet and Loris Karius.