The 33-year-old singer has revealed she and the Arctic Monkeys frontman were in talks about making a supergroup and even came up with the name, but she hasn't seen the 32-year-old musician ''for a long time''.
She said: ''We were meant to do a band together once, called Rupert The Band, me and Alex Turner.
''I haven't seen him for a long time. He lives in LA now.''
While she hasn't met up with Alex for a while, the 'Not Fair' hitmaker still listens to his dulcet tones and admitted she enjoyed the latest Arctic Monkeys album, 'Tranquility Base Hotel & Casino'.
She said: ''I like that new record. It's like cliché rock 'n' roll, but I like it.
''You know what I mean? I knew it wasn't going to be Sheffield rock again, but it's like everything that I expected from a bunch of lads living in Hollywood, from Sheffield, to come up with and I like it for that.''
Alex was clean shaven when he shot to fame with the 'Do I Wanna Know?' hitmakers - who dropped their debut album 'Whatever People Say I Am, That's What I'm Not' in January 2006, just seven months before Lily released her debut 'Alright, Still' - but he has now grown a beard, and Lily approves.
She said: ''It's a nice beard.''
Lily - who is a regular Twitter user - also admitted singers should probably follow in the footsteps of Adele, who she says ''hides and releases albums'', a lot more.
She added to NME: ''You never really here from Adele. She sort of hides and releases albums. I guess that's how it should be.''
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