Branthwaite, 20, went on loan to the Dutch club in the summer on a season-long loan and while dealing with injury, has still impressed for them.
He has made 12 appearances in all competitions and scored in the 4-3 win over current league leaders Feyenoord in September, but it had been speculated that Everton could recall him.
PSV Eindhoven themselves are third in the Eredivisie table, only three points behind the league leaders and they do not want things to change with Branthwaite.
The Eredivisie side are in talks with Everton to ensure the centre-back stays with them beyond the winter window, according to Dutch daily the Eindhovens Dagblad.
PSV are satisfied with the progress shown by the Everton loanee and do not want the Toffees to exercise any release clause.
Brantwaite, on his part, is also content with staying at the Dutch club, which could end up swaying Everton’s decision.
It remains to be seen if Everton agree to PSV Eindhoven’s proposal and let the defender stay at the Dutch club beyond the winter window.
