Liverpool captain Gerrard was critical of Mario Balotelli for stepping in front of Jordan Henderson to take a late penalty in Thursday's UEFA Europa League tie with Besiktas.
Balotelli scored to settle the game 1-0, but Gerrard labelled the striker "mischievous" and "disrespectful" for going against pre-match instructions.
However, Gerrard will have to muscle in on former Liverpool colleague and Galaxy captain Keane if he wishes to assume spot-kick duties.
When asked if Keane could relinquish his role, Arena answered: "He's taking them until he misses his next one. He's our captain and our penalty taker... until he tells me otherwise."
Gerrard will head to Los Angeles in July after his 17-year stint with his hometown club comes to an end, and Arena conceded surprise at the midfielder's excitement to join Galaxy, having met him this week.
"I saw him yesterday and I was surprised to see how excited Steven is to come to Los Angeles," Arena said.
"Obviously, he knows that he has a huge obligation to fulfill with Liverpool and to close out his career there in the right manner.
"They are involved in some cup competitions, and then he will be very eager to get to Los Angeles."
