That player is Micheal Keane, who according to Football Insider Everton are ready to sell this summer, while still doing everything possible to keep Branthwaite.
Keane has been effectively frozen out by Sean Dyche last season once he was replaced in the team by Branthwaite.
That came after a dreadful 4-0 defeat at Aston Villa two games into the campaign, in which he had made one of his characteristic costly mistakes.
Prior to that Keane had been a favourite with the manager who had re-united him with his former Burnley team-mate James Tarkowski at the heart of the Blues defence.
It did not work however, and after that car-crash match at Villa, Dyche simply had to make a change.
Bringing Branthwaite into his side was something many Evertonians had been pushing for and so finally the young centre-back got his chance.
And the defender - who came to be known as 'The Carlise Kaiser' - rose to the occasion and produced a string of superb performances week-in-week-out for most of the remainder of the campaign.
His pose, calm solid all-round defending and regular success going up against the best forwards in the Premier League, revealed how good a player he had become. He cemented a strong centre-back partnership with Tarkowski too.
Branthwaite's year-long loan in Holland with PSV Eindhoven the previous season made a significant difference to his confidence and consistency and he carried that on and even raised his game again for Everton.
Since the end of the campaign, there have been constant rumours that he will be leaving Goodison Park this summer, along with another top young talent Amadou Onana.
Manchester United were the club most strongly linked with Branthwaite and the Old Trafford outfit did eventually make a bid of around £35 million for a player Everton value at closer to £70 million.
That offer was rejected and so far United have not come back with another concrete bid while their attention seems to have switched to other targets.
Branthwaite himself has it seems decided to stay put on Merseyside, at least for now.
And, after some smart transfer business, the Blues financial position appears to be considerably stronger, making a quick player sale less necessary.
As for Keane he does not appear to have a long-term future at the Toffees, but selling him this summer seems less than straight forward a task.
For a start, who would put a decent fee on the table for a 31-year player who has become somewhat renowned for making basic errors and who has not been able to hold down a regular place in his current team, despite all it's struggles on the pitch?
Even if a buyer was found for Keane, Everton would then have to replace at least two reserve central defenders having recently sold Ben Godfrey to Atalanta. And that is without losing Branthwaite either, which is still very possible.
So, it certainly seems likely that the Toffees will be shopping for new centre-backs - as well as forwards and full-backs - in this summer's transfer window.