The pair have been in regular contact since the current Rangers boss took over from Gerrard in November last year following the ex-Liverpool midfielder's move to Villa Park to take up his first managerial job in the English Premier League.
Gerrard lost his job on Wednesday night after a 3-0 defeat to Fulham left his Villa side with just two wins and nine points from the opening league 11 matches. Van Bronckhorst expressed sympathy for his Ibrox predecessor and tipped him to return as an improved boss in his next managerial role.
"Since I joined here I have been in regular contact with Steven about his time at Aston Villa and the games we play at Rangers," the Dutchman said. "It's something that's part of your job. I sent him a message as well this morning.
"Eventually all managers will face it at one time and it's not what you want because we're all dedicated to our jobs. We give everything to get success to the club. But if that's not going well you know the pressure is on you and as an old player playing in these circumstances witht high pressure, I think he's familiar with that. For sure he will learn from his time at Aston Villa, and also from his time at Rangers, to be a better manager in his next job."