The Dutch giants signed Branthwaite from Everton in the summer on a loan deal and they have been pleased with what they have seen of the defender.
Branthwaite has been handed regular game time in Eindhoven, making 12 appearances in the Eredivisie and playing in Europe for the club.
Now PSV Eindhoven are seeing if they can sign him permanently and are speaking to Everton to do so, according to Dutch daily the Eindhovens Dagblad.
The Toffees are trying to strengthen this month, but are walking a tight financial tightrope, even though they are set to bank £45m from selling Anthony Gordon.
Cashing in on Branthwaite may be appealing to the Premier League club as it could give incoming boss Sean Dyche further scope for additions.
And PSV Eindhoven will see whether there is common ground for a deal to be struck.
Branthwaite, 20, is under contract at Everton until the summer of 2025 and has been regarded as a bright prospect.