The Dutch newspaper report that the Manchester United legend and Championship club are 'virtually agreed' on a 'multi-year contract'.
Van Nistelrooy bid farewell to PSV over a year ago. He spent a season there on his first job in management, winning the Dutch Cup and Dutch Super Cup.
Under the former striker, PSV finished second in the Eredivisie in the 2022/23 season and qualified for the Champions League.
Van Nistelrooy stepped down because he felt a lack of support from within the club.
Burnley, where Netherlands striker Wout Weghorst is currently under contract, were relegated from the Premier League last season - under Kompany - after finishing 19th.