Scouts from all three clubs have been sent to watch the 17-year-old, following his impressive start to the 2019/20 campaign, but the Foxes are believed to be firmly leading the hunt.
According to Football Insider, Brendan Rodgers is confident in signing the young star and Leicester are expected to make their interest official in the beginning of next year.
The Foxes have scouted him regularly, regarding him as a potential long-term option for their back line. The centre back only made his debut for Carlisle this season, first starring in their EFL Trophy meeting with Blackpool in October.
Overall, the commanding defender has made 13 appearances in all competitions for Carlisle this campaign. He also became their youngest ever goalscorer when he found the back of the net against Morecambe in the EFL Trophy in November.
While he recently signed a two-year deal with his current club, it hasn't halted any intense speculation regarding a potential move away from Brunton Park, and a move to the Premier League is looking likely.
Branthwaite, who turns 18 next June, has worked his way up through the ranks – joining the club at just eight-years-old and finally establishing himself as a senior player this season.
As well as Branthwaite, the Foxes are also reportedly eyeing up Royal Antwerp right-back Aurelio Buta for the next transfer window. The defender has also been linked to other Premier League clubs including Wolves, Watford and Southampton.
Leicester currently sit second in the table, 10 points behind leaders Liverpool. The Foxes are having a fantastic season first full season under Rodgers, but the Northern Irish coach seems keen to strengthen his squad at the next given opportunity.