The Clarets approached their npower Championship rivals for permission to speak to Lambert. But Norwich refused and Lambert insisted he is staying at Carrow Road, leaving Kilby to look elsewhere for a new manager.
The Turf Moor supremo said: "I would like to stress to supporters that Paul Lambert was only one of a number of targets we would like to interview for the post of Burnley manager.
"There are other targets, but someone had to be the first and Paul was already in employment, so we acted through the proper channels and made a formal enquiry to speak to Paul.
"Obviously this was refused and we respect Paul and Norwich's decision and now we move on to the other persons on the shortlist.
"There were half a dozen names on our shortlist and the task starts again tomorrow (Sunday). I do think the situation needs to be resolved as quickly as we can, but we cannot let that be the overriding factor. It is about appointing the right man for the job to take this club forward."