Burnley chairman Barry Kilby was upset with Bolton and claimed they were "not the best of pals", claiming the Glasgow-born boss had taken "a sideways step".
With the clubs due to meet on January 26 it promises to be an intriguing situation on the pitch - and in the boardroom.
Asked about Bolton's relationship with Burnley, Gartside said: "It is the same as it has ever been - we are competitors in this business.
"No doubt we will sit down and have a drink again in a couple of weeks and get on with life.
"A sideways step? That is not for me to judge. I do not know what Burnley are like. I can only judge what we have here and Owen likes what he sees."
Coyle has replaced Gary Megson, who failed to win over a section of the support despite keeping the club in the top flight.
The former St Johnstone manager is keen to get the fans on board and reflected on the time when he was a Bolton player in the 1990s.
Coyle said: "The fans played such a special part in that era. Phil Brown was the captain for a couple of years and we had some terrific players.
"There was John McGinlay, Alan Thompson, Jason McAteer, Alan Stubbs and Tony Kelly to name but a few.
"The was a special rapport within the squad. We had a fantastic team spirit, which was evident and the fans could see that.
"That is something I am looking to foster, like we did at Burnley.
"I want to make sure that the fans know when the players wear that shirt they wear it with pride and give everything in terms of trying to win a match."
Coyle has encouraged his teams to play an expansive game and he knows that is what Bolton supporters want after Megson's departure.
However, he said: "I don't think you can change from one style to another overnight but we will look for gradual progression.
"I believe we have the players to keep us in the Premier League - and that has to be the priority.
"It will involve a lot of hard work. There is no magic wand where you sprinkle fairy dust and everything happens."
Coyle is now looking forward to a home game with Arsenal on Sunday and added: "It cannot come quickly enough for me. I know we will have to give our very best if we are to get something from the match."
Arsenal 4/7, Draw 11/4, Bolton 11/2