The band - which also includes Miles Kane, Jet's Nic Cester, Jamie Davis, and Sean Payne of the Zutons - started off as a "bit of fun" and Muse star Matt admitted he's enjoyed not having the "pressure" to write new music.
However, they haven't ruled out the possibility of making their own material in the future.
Blur guitarist Graham told The Sun newspaper: "I’ve thought about how it would go and what direction it would be. I’d be up for putting some demos forward.”
And Matt - who is actively working on new stuff for Muse's follow-up to 2018's 'Simulation Theory' - added: “Writing our own songs could happen one day.
“But at the moment we’ve enjoyed having no pressure to do that kind of stuff. We’ve enjoyed treating it more like the way DJs put together a set. It’s a different approach and one thing that sets us apart in a way.
“We could go down the road of doing songs but I personally like the idea of digging up music from the past. I’ve started writing a new Muse album as well.
“I started writing when all the protests and all the chaos were kicking off. It’s a weird thing to say but it suits the music I write. It keeps you a little bit on edge.”
The supergroup have played a select number of gigs, including a house party, Stella Mccartney's fashion show and the Teenage Cancer Trust at The Royal Albert Hall.
The Jaded Hearts started out covering Beatles songs and have recently expanded their repertoire and released a record of covers of "lost" greats, including Northern Soul and Motown tracks, on the LP 'You've Always Been Here'.
Matt recalled the time Beatles legend Sir Paul McCartney joined them on stage for a cover of The Beatles hit 'Helter Skelter' at his daughter Stella's show in Los Angeles in 2018, and said it was like being in the 'Get Back' group.
He recalled: “It wasn’t planned, (it was) totally random.
“Paul just jumped on stage and started singing with us. For that one moment, we thought, ‘Hold on a minute . . . we are The Beatles!’”
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