Coyle signed Gunners midfielder Jack Wilshere on loan last January and has been linked with a similar move for Vela, who the Scot described as an "outstanding" player.
"It's fair to say I would like to help the group that we have," Coyle said. "We had only 15 players at the weekend at Anfield, so we'd like to get one or two in to help the team. Time will tell who becomes available and who wants to come.
"Carlos Vela is an outstanding player. I think what happened was that (Arsenal manager) Arsene Wenger said he might consider letting him go out on loan and then people have looked and put two and two together.
"Bolton have a very good relationship with Arsenal, obviously with Jack Wilshere last year - hopefully we played a small part in his continuing development.
"But there is no doubt that Carlos Vela is a wonderful player."