Aghayere was released at the end of last season by the Clarets to bring to an end three years at Turf Moor.
The Madrid-born keeper arrived at Burnley as a scholar ahead of the 2014/15 season and turned in some impressive performances in the youth team.
He spent the second half of the 2015/16 season on loan with Evo-Stik League North side Colne FC after which he was given a third year as part of his scholarship.
He saw little action for Burnley last season during which he returned to Colne for the full season, after which he was one of the young players released.
Now aged 19, Aghayere has signed up with the Ammies.