The England international, who made 65 league appearances in the previous two seasons, has been sidelined since August.
The 23-year-old signed a new deal in July that will keep him at Everton until 2030.
"Following specialist consultation, it has been determined the defender will undergo a surgical procedure tomorrow to address the issue," Everton said in a statement on Monday.
"Branthwaite will commence a structured rehabilitation programme under the supervision of the club's medical team following the operation."
Everton, who have 11 points after eight league games, host Tottenham Hotspur on Sunday.