The "All I Want For Christmas" singer hit up Catch Steak in the popular celeb vacation destination Sunday night with Anderson ... and there were signs of a potential new couple.

Mariah and the Silk Sonic scribe were holding hands as they headed into the fancy steakhouse ... and they both appeared super happy, just look at the grins and smiles here.

Anderson helped hold the door open for Mariah ... and he kept her close with one arm around her waist.

See, ladies ... chivalry ain't dead yet!!!

Even though they locked hands on their way into dinner, folks who were there say Mariah and Anderson left the restaurant separately ... but not before sparking romance rumors.

However, our sources tell us Mariah and Anderson are NOT dating.

Instead, we're told the singers are working on new music together and have been in the studio in Aspen ... taking a break from their work to grab dinner at Catch Steak.

Mariah and Anderson are both single, though ... as we first told you, earlier this year he filed for divorce from his wife.