The 29-year-old centre-back has signed a pre-contract and will join the club officially on a two-year deal at the start of July when his contract with Wycombe Wanderers expires.
Right-footed Stewart has been a key player for the League One club in recent seasons, helping Wycombe win promotion to the Championship in the 2019/20 campaign.
Across two spells he has played nearly 300 games for the Chairboys in the second, third and fourth tier of English football. He also had a period with Crewe Alexandra.
The experienced Stewart joins fellow defenders Liam Scales and Jayden Richardson in signing with the Dons this week, with Albanian international midfielder Ylber Ramadani the first arrival of the summer,
“I’m delighted to get Anthony over the line,” Jim Goodwin told the club’s website. “We’ve been working on this one for quite a period of time now.
“Anthony has played a lot of games in League One and the Championship in England. He is a proper, old-fashioned centre back that enjoys the art of defending.
“We’ve got some exciting young players at the club, and I think it is always important that you have some experienced players throughout the team and at 29, Anthony will be one of the more senior figures within the group.
“We look forward to working with Anthony and I’m sure he’ll have a positive impact in the season ahead.”
Aberdeen have returned to pre-season and will travel to Spain for a training camp next week.