And the fact that Bilic, who enjoyed a successful two-year spell as Besiktas boss before returning to East London, is in charge of the Hammers could put them in pole position over Arsene Wenger's men.
Ozayakup actually had a spell in the Arsenal first-team squad in the 2011-12 season after graduating from the academy.
But he only made one senior appearance against Shrewsbury Town in the League Cup and has mixed memories of his time at the club.
He set up Yossi Benayoun for a goal in a 3-1 win in that match against the Shrews.
But he was also on the bench for Arsenal's 8-2 massacre at the hands of Manchester United and then again watched on as a substitute when Wenger's men beat AC Milan 3-0 in the Champions League, but went out 4-3 on aggregate.
Big talent, big price tag
According to Fanatik, the Turkey international has caught the eye of several top clubs with his fine performances this season and could cost around £14.7 million.
He may be an exciting player that Bilic knows well, but Hammers co-owner David Sullivan recently warned any new signings in January would only be possible if players are sold, as reported by the Evening Standard.
Bilic could find himself with a midfield headache with everyone fit.
He currently has Mark Noble, Cheikhou Kouyate, Dimitri Payet, Manuel Lanzini, Victor Moses, Michail Antonio, Reece Oxford and Pedro Obiang competing for places, while Alex Song will return from injury in the next couple of weeks to bolster his numbers further still.
The former Croatia boss signed 12 players over the summer, but may want to freshen things up in January depending on the club's form between now and then.
He has already cleared some space in the squad and on the wage bill after midfield outcast Morgan Amalfitano was this week released by mutual consent from a two-year contract he signed back in March, as revealed by the club's official website.
But whether he is willing, or able to sell any of his fringe players to fund a move for Ozyakup remains to be seen.
Joined Arsenal academy from AZ Alkmaar in 2008.
Signed by Besiktas in 2012 for around £350,000 and has made 85 appearances scoring 11 goals.
Can play as both a defensive and offensive midfielder.
Represented Holland at Under 17 and Under 19 level, scoring seven goals in 24 games before switching allegiance to Turkey, since when he has played six times for the Under 21s and 12 times for the senior side.