The Dutch giants have had a disastrous season and want a coach who can take over long term.
John van ’t Schip is interim boss until the end of the season but not likely to get the job.
A Ten Hag return is not currently an option after the FA Cup win over Liverpool gave him renewed hope of staying at United.
So Van der Gaag is being considered and his work alongside ETH has not gone unnoticed.
The pair linked up for a year in Amsterdam before moving together to Old Trafford in summer 2022.
It would be a blow for the United boss to lose his No 2, particularly after coach Eric Ramsay left last month to become boss at MLS club Minnesota.
Ramsay had been on the United coaching staff for almost three years and also worked under Ole Gunnar Solskjaer, Michael Carrick and Ralf Rangnick.
Van der Gaag has previously managed in Portugal, Cyprus and his native Netherlands before becoming Ten Hag’s Ajax assistant in 2021.
However, he has not taken charge of a senior side since leaving NAC Breda in 2019.
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Ajax are currently fifth in the Eredivisie after recovering from a slow start to the season which saw the club flirting with relegation.
They remain 31 points adrift of league leaders PSV, though, and are unlikely to secure a Champions League return after also missing out on qualification last term.
Aston Villa dumped the four-time European champions out of the Europa Conference League earlier this month.
Meanwhile, they were knocked out of the Dutch Cup by fourth tier amateur side USV Hercules in December.