At one stage yesterday Neil moved to favourite for the job, above Chris Hughton and Paul Cook who have been strongly backed in recent days.
There is not thought to have been any approach from the Robins to North End to speak with Neil, the speculation at the moment all coming from the bookies’ odds.
Those can change quickly, it only needing a few bets placed for someone to rise rapidly in the betting.
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However, Neil is well thought of in the game with a promotion to the Premier League on his CV.
Three times he has taken Preston to within sight of the play-offs, only for them to miss out on a top-six finish. North End have of course been here twice before.
West Bromwich Albion tried to get him in April 2019, with Stoke doing likewise last October.
Off the back of the interest from the Baggies, Neil signed a new contract at Deepdale.
Any club wanting him would have to pay out the remainder of that contract as compensation – he has got two years of it to run.
Bristol City sacked head coach Lee Johnson early in July and put his assistant Dean Holden in interim charge for the final five games.
The fifth of those was against North End last week, that a 1-1 draw at Ashton Gate.
Holden is being considered for the job, with Hughton well thought of having already been interviewed.
Whether they would be tempted to go for someone already in a job and have to pay compensation, remains to be seen.
Neil and his family are settled in Preston and he enjoys his job at PNE, even if the budget restraints have at times frustrated him.
At 39, he is still relatively young as a manager even though this is his third job.
He started off at Hamilton, the job he did there in guiding them to the SPL landing him the Norwich role.
With pre-season training starting on August 13, PNE will not want any speculation dragging on.