Brendan Rodgers is expected to become Leicester City’s manager after Celtic gave him permission to speak to the Premier League club.
The Foxes are keen to install a successor to the sacked Claude Puel as soon as possible.
First-team coach John Kennedy is set to take charge of Celtic for Wednesday’s Scottish Premiership trip to Hearts.
“Brendan has indicated that this is an opportunity he wishes to investigate further,” Celtic said in a statement.
Former Liverpool boss Rodgers cut short a BBC interview when asked about the Leicester vacancy after the champions’ win over Motherwell on Sunday.
They are eight points clear of nearest rivals Rangers in the league as they chase an eighth successive title.
Rodgers has won two Scottish Premierships, three Scottish League Cups and two Scottish Cups since arriving at Parkhead in the summer of 2016.
They reached the group stages of the Champions League in his first two seasons, but were knocked out in the third qualifying round by AEK Athens this term amid unrest between Rodgers and the Celtic board over a failure to strengthen the squad last summer.
Last week, they bowed out of the Europa League at the last-32 stage for the second successive season, losing to Valencia.
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‘It is no surprise other clubs want him’ – Gerrard
Rangers boss Steven Gerrard played under Rodgers at Liverpool, where he narrowly failed to guide them to the Premier League title in 2014 before his departure in October 2015.
Gerrard, speaking on Tuesday, said: “Brendan has done ever so well at Celtic over the last few years. It is no surprise other clubs watch him and want to acquire his services.
“I have worked with Brendan, he is a very good coach. I am not surprised at the timing. From our point of view, we will keep an eye on it and see how it pans out. My focus is on Rangers.”
Puel was dismissed after 16 months in charge following Saturday’s 4-1 defeat by Crystal Palace.
The 57-year-old Frenchman left the club 12th in the Premier League having lost five of their past six league games.
First-team coaches Mike Stowell and Adam Sadler were put in temporary charge. Leicester play Brighton at home in the Premier League on Tuesday (19:45 GMT).
‘The news Celtic have been dreading’ – analysis
BBC Scotland sports news correspondent Chris McLaughlin
It’s the news the Celtic fans had been dreading. Since taking over in 2016 he has delivered everything domestically – the double treble was secured at the end of last season and the club were on the way to making it three in a row.
But there was always a feeling the former Liverpool boss wanted to try his luck once again in English football’s top flight. Suggestions he wasn’t getting the boardroom backing he was looking for may also have hastened his exit – but it leaves Celtic looking to replace one of their most successful managers.
BBC chief football writer Phil McNulty
Rodgers has rebuilt his reputation in Scotland after his ultimate failings at Liverpool and is certainly not short of personality or charisma.
The 46-year-old also plays attractive, attacking football and shares in Puel’s penchant for promoting young talent. And after three years and seven trophies – to date – maybe Rodgers will feel that the time is right to leave.