The Daily Mail have claimed that Middlesbrough, Sheffield United, Huddersfield Town, Blackpool, Cardiff City and Sunderland are all keen to sign the 19-year-old on loan with PSV Eindhoven and Schalke also eyeing a move for the youngster.
The defender has played 13 teams for Everton since joining the Toffees from Carlisle United two years ago. He made six appearances for the club in the Premier League last term, scoring one goal which came in Everton's 1-1 draw at Chelsea in December.
He was sent off in the club's penultimate home game against Brentford in May and played in the final game of the season, a 5-1 defeat at Arsenal.
Branthwaite has experience of the Championship after making 10 appearances for Blackburn Rovers during a six-month loan spell in the second half of the 2020-21 campaign.
Huddersfield have made one signing so far this window with Will Boyle joining on a free transfer from Cheltenham Town. More incomings are to be expected with 11 players leaving the club this summer, including those who have been released and those going out on loan.
Middlesbrough made three signings last week with Liam Roberts, Ryan Giles and Darragh Lenihan all arriving at the club and are expected to have a busy summer as they target promotion under Chris Wilder.
Sheffield United are yet to announce any signings this window after releasing David McGoldrick, Lys Mousset and Luke Freeman at the end of the season.
The three Yorkshire clubs could be an attractive proposition for Branthwaite, who is out of contract at Everton next summer, with the trio eyeing involvment in the Championship promotion battle next term.