Kilby is hoping to appoint a successor to Coyle - who has been officially unveiled at the Reebok Stadium - before Burnley's trip to Manchester United on Saturday.
Reports have suggested Huddersfield boss Lee Clark was interviewed by Kilby and operational director Brendan Flood on Sunday night, while Doncaster's Sean O'Driscoll and former Sheffield Wednesday manager Brian Laws are also thought to have been quizzed about the position.
Burnley have confirmed interviews have been "ongoing" - although they would not be drawn on the names of any candidates - and, in a statement on the club's website, Kilby reaffirmed his optimism and determination to find the right man to take things forward.
"It's been a turbulent few days in the history of Burnley Football Club," Kilby said.
"But sometimes these things happen for a purpose, so let's all pull together and work as one for the betterment of this.
"When I think back to November 2007 and the appointment of Owen, I remember feeling good about the future.
"I have that same feeling again now as we concentrate on replacing him.
"We have a clean sheet of paper and a reputation within the game that has been enhanced by our promotion.
"We intend to try to get somebody who will be right for us and keep us in this Premier League."
Kilby stressed his search for a new man would not compromise the club's financial position.
"Ultimately, we did all we could to keep Owen here as our manager and that was not enough," Kilby said.
"But I am not going to throw away all we have achieved by offering up funds we simply cannot afford.
"You can only work within the budgets you set out at the start of the season and there is absolutely no way I want to end up like some other illustrious clubs who have, dare I say it, bet the ranch."
Clark, the former Newcastle midfielder, has emerged as the new favourite to take the Turf Moor hotseat.
The 37-year-old only moved into management in late 2008 but his backroom staff at the Galpharm Stadium have a wealth of experience.
Clark's assistant, Terry McDermott, served in the senior coaching team at Newcastle for nine years across two spells, working for Kevin Keegan, Kenny Dalglish, Glenn Roeder and Sam Allardyce.
Derek Fazackerley, Clark's first-team coach, has also worked at Newcastle, was England's assistant manager under Keegan and, more recently, was an advisor to Sven-Goran Eriksson during his one season in charge of Manchester City.
Clark also employs former Burnley loan star Andrew Cole as a striker coach two days a week.
Simon Grayson is another name to be linked with the post and the Leeds boss was cautious when asked about the subject after his team's 1-1 draw against Wycombe on Saturday.
"I'd be flattered," Grayson said.
"But, until I'm told there has been an approach, I'm 100% committed to my job here.
"We all want to manage in the Premier League but my ideal scenario is to manage this club in the Premier League."
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