Coyle has won many admirers since guiding Burnley into the Premier League last May and putting the likes of Manchester United, Everton and Sunderland to the sword at Turf Moor this season.
Coyle's commitment to attractive, free-flowing football has never wavered and Branagan feels that will not change if he is installed at Bolton.
Branagan, 43, said: "I've followed Owen's management career very closely, first in Scotland and then since he went to Burnley in November 2007.
"I wrote to him when he was appointed at Burnley and wished him all the very best.
"It's been no surprise to see him do so well and if he gets the Bolton job then he would be an outstanding choice.
"Owen has got so much energy, enthusiasm and desire. He had it as a player at Bolton and he's taken it into management.
"Just look at the progress Burnley have made since Owen went there - they always try and play football the right way.
"I would expect Owen to adopt the same style of play at Bolton because I actually think he will inherit a very good group of players.
"The likes of Matty Taylor and Gary Cahill are fantastic footballers and I think Owen will give Bolton a different approach, rather than using Kevin Davies as the main target all the time."
Bolton have generally struggled since Sam Allardyce walked out in April 2007.
Sammy Lee, Allardyce's successor, ripped up the blueprint which had harvested so much success under Allardyce and tried to make the Trotters a more attractive side.
His plan had disastrous consequences and he was sacked with the club bottom of the table with a pitiful five points from 10 games.
Megson attempted to revert to the more direct approach employed by Allardyce but he never won over a sceptical Bolton public.
Coyle, in contrast, already enjoys cult status among Trotters fans for his part in their promotion to the top flight almost 16 years ago.
In the 1995 play-off final against Reading at Wembley, Bolton were 2-0 down and staring at defeat.
Royals striker Stuart Lovell had the chance to put his side 3-0 up from the penalty spot but Branagan saved his effort and Bolton roared back to draw level with goals from Coyle and Fabien De Freitas.
A tense period of extra-time saw the Trotters eventually win 4-3 and Branagan, who now works as a teacher in Bolton, added: "It was a great time to be at Bolton and Owen and the rest of the team will always be remembered very warmly by the fans.
"As a club we were going from strength to strength and I would have liked Owen to stay longer than he actually did.
"He was a great character. He was, and still is, tee-total.
"He would always be drinking Irn Bru - and lots of it - but never touched a drop of alcohol.
"Quite often he would be the last person to leave after training and would stay playing head tennis with the young apprentices at Burnden Park.
"He's just a great guy and I would be delighted to see him appointed as manager of Bolton Wanderers."
Arsenal 1/5, Draw 5/1, Bolton 14/1