Wanderers travel to Merseyside on Wednesday night to take on Everton, where Moyes has been in charge for more than eight years.
His achievements over that period - which include guiding the Toffees to a fourth-placed Premier League finish in 2005 and the FA Cup final in 2009 - have seen him emerge as one of the most frequently-touted candidates to succeed Sir Alex Ferguson at United.
That has come as no surprise to Coyle, who feels Moyes' exploits warrant attention from the game's biggest clubs.
"It certainly wouldn't surprise me if one of the elite clubs in world football had been looking at David Moyes to be their manager," Coyle said.
"It's been mentioned before about him being a successor to Sir Alex Ferguson and that doesn't surprise me at all because I know how hard he works and how good he is at his job.
"If he is given the right tools, there is no doubt he would go on to win trophies."
Coyle could see the management potential in Moyes, his fellow Scot and old friend, from an early stage.
"Fantastic doesn't even do him justice," Coyle said. "I know him from when I was at Bolton as a player - Moyesy was at Preston and John McGinlay, who I played with, was pally with him from Shrewsbury.
"We used to meet up and I knew how much he loved the game - and from a coaching perspective, even at that stage when he was playing.
"So it didn't surprise me when he went on to be, as he is right now, an outstanding manager who has done a remarkable job.
"Year after year Everton continue to get stronger and stronger."
Coyle had to make the journey to Goodison Park twice last season in the Barclays Premier League, first with Burnley and then Bolton having moved from one club to the other in January.
He was defeated 2-0 on both occasions and is expecting another testing time on Wednesday.
"I remember the Everton game very clearly - we (Bolton) were terrific at Everton and very comfortable in the game," Coyle said.
"Then we had a player sent off and Mikel Arteta moved a free-kick a number of yards and curled one in the corner.
"I remember that feeling of disappointment because we did ever so well, but that is how tough it is at Everton.
"I went there with Burnley as well last year and it was 0-0, before we had a man sent off as well and lost a goal.
"So we need to be concentrated and focused for the duration of the match, knowing that Everton have quality."