The former Netherlands international is currently working at Guangzhou R&F, having moved to the Chinese Super League club in January.
Van Bronckhorst previously spent four years in the top job at Feyenoord, leading the club where he began and finished his playing career to the Eredivisie title in the 2016-2017 season.
While happy with his current employers, his aim is to work in the Premier League one day.
A return to take charge of the Gunners, another former team from his playing days, is a "big dream" for the Dutchman.
"At the moment I'm enjoying my work in China," Van Bronckhorst said.
"But as a player I had ambition and as a coach I have ambition. The Premier League for me is the number one league in the world to be working at and, for me, it is definitely an ambition to work in.
"I always said if I could coach the teams I played for it would be a big dream."
After his tenure at Feyenood came to an end, Van Bronckhorst took the opportunity to visit Manchester City and to spend time watching Pep Guardiola at work.
"Last year I spent five months with the City group and a lot of time in Manchester with Pep," Van Bronckhorst said.
"He was very open. He talked about the training sessions and I could watch and see everything and talk with his staff."