Former Dagenham and Dartford defender Arber, whose father Bobby has also worked for the Gunners as a scout, posted a Tweet on his personal account, @1Arbs, on Monday that suggested he had bet on Germany midfielder Ozil's impending move to the Emirates Stadium.
"Lovely bit of 14-1 Mesut," he wrote.
The Premier League club are now looking into the potential breach of gambling rules.
A club spokesman said: "We take this very seriously.
"All Arsenal employees and workers are well aware of their responsibilities in this respect and we are looking into this as a matter of urgency."
If Arber had bet of the arrival of the 24-year-old, who completed a club-record £43million move from Real Madrid late on transfer deadline day, the 35-year-old is likely to have broken terms highlighted in the Football Association's guide to betting rules.
The rules state: "You are not allowed to place a bet on a game or competition in which you have any influence, either direct or indirect.
"The above includes all bets related to the following: Any other events involving your club or other clubs playing in the same league competition (such as next manager markets)."