Coyle signed Gunners' midfielder Jack Wilshere on loan last January and has been linked with a similar move for Vela, who the Scot has 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."
He added: "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."
Coyle was less forthcoming when it was suggested to him that FC Utrecht's Michael Silberbauer might be another potential transfer target, but he did stress that there had been no enquiries about Cahill, despite reports that Chelsea are set to bid for the England defender.
Asked about Silberbauer, Coyle said: "I think we are all aware about those types of players, but it's the usual names being bandied about.
"I think there was speculation regarding Gary Cahill on Sunday - I can sit here now very comfortably and say I have had no dialogue with any other football club regarding Gary Cahill or any of my other players.
"Long may that continue, because I think it is important that the group we have got, we keep them together and we try and add one or two fresh faces to help what we have."