Babalj travelled to the Netherlands to undergo a medical and resolve personal terms after AZ scouts visited Melbourne this year to watch him play at Heart.
The striker's potential was recognised last year as Serbian Superliga side Red Star Belgrade moved to sign Babalj, where he tasted 2012-13 UEFA Europa League action.
Babalj returned to Heart in January this year for the remainder of the 2012-13 season, signing a deal for the 2013-14 campaign.
However, both parties agreed the striker could travel overseas during the off-season.
Heart chief Scott Munn said: "When Eli returned, we sat down and agreed that he could explore his options in Europe with the view to sign with a club if the opportunity presented itself.
"We didn't want either Eli's or the club's pre-season to be disrupted so we allowed him to travel during the off-season.
"Eli has attracted attention from Europe which is testament to his calibre and character and we wish him all the best at AZ."