The Daily Mail says the 17-year-old, a left-sided centre-back who is already in Carlisle's first team, is expected to be sold next month amid interest from a number of Premier League sides.
The starlet has made 10 appearances for the League Two club this season.
After playing in Carlisle's FA Cup first round win over Dulwich Hamlet, Branthwaite told the club website: "That was a real experience.
"I had some butterflies throughout the game, but that's normal I suppose. I just got on with it.
"Playing in the leagues and cups is really good for me at my age. It's helped my confidence a lot, so I've got to keep pushing on."
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