Coyle led Burnley into the top flight for the first time in 33 years after moving from St Johnstone in November 2007, but believes Bolton is a better option.
Burnley graciously thanked Coyle for his achievements after confirming that a compensation package had been agreed with Bolton.
"The officials of Burnley Football Club would like to place on record their thanks to Owen for his services to the club and contribution to our success," read a statement on the club's website.
Gartside later admitted on Sky Sports News: "It's been a long, drawn-out week.
"Owen was the only one on the list. I'd got a list of 30 names but the only one we wanted was Owen. We haven't interviewed anyone, we haven't contacted anyone. It was just Owen that we wanted."
As for what makes Coyle special, Gartside added: "It's his desire. He had to come back here, really.
"He's well known to the place, he came through an era which was special to everybody, including me. Bruce Rioch revitalised Bolton in the early 90s. That three-year period was special and Owen was part of that.
"He is infectious. He still is infectious, even more so than before."