The Germany midfielder moved to the Emirates from Real Madrid for a club record £42.4m fee on transfer deadline day.
Arber wrote on his Twitter feed: "Lovely bit of 14-1 Mesut."
An Arsenal spokesman said: "All our employees and workers are well aware of their responsibilities in this respect and we are looking into this as a matter of urgency."
The FA regulations 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)."