Burnley officials insisted that Coyle had to leave to catch a scheduled flight at Luton airport to see his family in Scotland and assistant manager Sandy Stewart was left to field questions.
Stewart said: "There is nothing to read into this.
"Owen had to get away to see his family to celebrate the New Year and if he had done the press there is a real danger he would have missed his flight."
Burnley confirmed that, as of 2.30pm on Saturday afternoon, Bolton had still not been in touch and Stewart is in no doubt Coyle is happy at Turf Moor.
The former Bolton striker was also heavily linked with a move to Celtic in the summer before opting to stay in the north west and Stewart believes the outcome will be the same this time around.
He said: "It's simply speculation and we had it in the summer with Celtic.
"Until something else happens, I can't make much comment but what I do know is he is very happy at Burnley.
"We have worked hard and we think we have done a good job here to put Burnley in good standing.
"We are happy here, we think we have taken the club forward and that is all we can concentrate on.
"It is up to other people to decide whether it would be a sideways move. If that moment in time comes when Owen has to decide, then he might decide that it is.
"If you asked every player in that changing room, they would hope that it was not Owen Coyle's last game in charge and they don't think it is."
Coyle made a point of approaching the away end after the final whistle and applauding them for their support.
Goals from Graham Alexander and Steven Fletcher had put the Clarets in control but Paul Ince's side mounted a fierce late surge that culminated in a goal from Dean Morgan.
Stewart insisted Coyle was not saying goodbye to the fans who had made the journey - but did admit he hopes to continue to partner Coyle wherever he ends up managing.
He said: "It was not a wave goodbye. He does the same after every game as he appreciates the support when they have made a journey.
"Right now there is nowhere for him to go so he will stay at Burnley.
"We have a great partnership. We have worked together at St Johnstone and Burnley so, whatever happens, hopefully that partnership will continue.
"At this moment in time, that will be at Burnley."