The 20-year-old was released by Villa in May after his contract expired having joined the Midlands club from Watford as a youth player in 2007.
Forrester, capped at England Under-17 level, did not make a senior appearance for Villa but was a regular performer for their Academy and reserve sides.
He also spent part of last season on loan at Scottish Premier League club Kilmarnock, notching up seven appearances.
The attacking midfielder is currently undergoing a trial period with the Eredivisie giants in hope of earning a contract.
And De Boer is hopeful of bringing the youngster to the Amsterdam Arena on a permanent basis after witnessing him in action for Villa reserves last year.
He told De Telegraaf: "He is close to a deal.
"He's a good player. But it's not only about the two per cent of the time you have got the ball during a game. The 98% you don't have the ball is more important. And I think he has to develop that side of the game.
"Harry will go with Jong Ajax (Ajax's reserve squad) to a training camp in (Dutch region) Zeeland. After that we will make a decision. But it surely looks good."