Coyle steered the Clarets into the top flight via the Championship play-offs but has since been linked with a move away from the club.
Celtic are understood to be interested in appointing Coyle and the speculation has yet to die down despite the manager saying he is focused on his current job.
Alexander hopes he will stay at Burnley and admits it would be a big blow if the Hoops were able to lure Coyle to Scotland.
"I don't know for definite but it is going on a little bit longer than we first thought," said the veteran defender.
"Every player at Burnley is hoping he stays. He has been fantastic since he walked through the door and obviously we don't want to lose our manager.
"He is a great manager. He took a team that doesn't have a big squad or a lot of money to spend and got us into a tight group.
"We have achieved great strides this season and he is the leader of the pack."
Scottish veteran Alexander is convinced that Coyle would be good enough to succeed at Parkhead.
"I think he could (do the job) but I don't want to talk him up too much in case that makes their mind up," he added.