Gerrard is out of contract with the Galaxy following the completion of the Los Angeles club's season
Gerrard is out of contract with the Galaxy following the completion of the Los Angeles club's season last weekend in the MLS playoffs.
The 36-year-old has been linked to a return to Liverpool in a coaching role while other reports in Britain have suggested he could be tempted into extending his playing career with Celtic.
Arena said no decision on Gerrard's MLS future had been taken yet, but said he hoped the midfielder, and Irish striker Robbie Keane, who is also out of contract, could be tempted to stay with the Galaxy.
"We were honored to have Stevie here as well as Robbie and are hopeful they continue to be part of our club but only time will tell," Arena told reporters, saying he had not discussed future plans with either player.
"I haven't spoken to those players. We need to let them have time to comment on what they want to do in their future rather than me commenting on it now," Arena told AFP.
"The season just ended. We need a little time to reflect. I'm sure both Stevie and Robbie at the proper time will discuss their futures."
Arena, meanwhile, paid tribute to both Gerrard and Keane's contribution at the club, praising Gerrard for slotting a penalty in Sunday's shootout defeat to Colorado after a lengthy injury layoff.
"If Sunday was his last game, to finish the way he finished with that penalty and the 45 minutes he played after being out for close to two months, was pretty impressive," Arena said.
"With Robbie Keane, there are very few players who have contributed as Robbie has," he added.
"Six unbelievable years, three championships, more than 100 goals. He embraced the league.
"You don't always see that from international players."
Arena said while the Galaxy envisioned changes to their 2017 roster, reports suggesting stars such as Wayne Rooney and Zlatan Ibrahimovic could be California-bound were wide of the mark.
"Are (Cristiano) Ronaldo and (Lionel) Messi in those discussions as well? Let me give you a little hint -- maybe 90% of the things that you see in the papers are not accurate," he said.
"There are going to be changes. We can get better, for sure. But all the things we're going to do with our team we don't discuss publicly. If you're gambling you don't turn your cards over to show everyone."