Amid fresh interest from Manchester United and Real Madrid ahead of the summer market, the young defender is intimating a willingness to consider extending his stay at Goodison Park.
Sky Sports says Branthwaite is prepared to sit down and discuss a new contract with Everton chiefs.
While the Toffees are unlikely to be able to match rival offers, Branthwaite will hear them out after being encouraged by the team's revival under manager David Moyes.
"Jarrad is a top player,” said Moyes last week. “If people want to buy Jarrad then I've said many times, bring a big bag of money.
“Jarrad has done really well. I think he could do better. There are still moments at times when he plays like a young centre-back, but I think he's got an awful lot going for him.
“He’s already been involved in the England squad before I came here…and I think he's got a good mentor beside him in Tarky (James Tarkowski). That'll be a help in improving him."