The Hoops have been looking for a new boss since Neil Lennon resigned in February, before Old Firm rivals Rangers stopped them winning a 10th successive Scottish Premiership title.
Howe left his role as Bournemouth manager by mutual consent last August, after being unable to keep the Cherries in the Premier League, having previously guided them to the English top flight from the bottom of League Two across two seperate spells.
The 43-year-old is now the clear favourite to take over at Celtic with negotations reported to have reach an ‘advanced stage’ following talks with members of the club hierarchy, including majority shareholder Dermot Desmond.
The Sun also claim that incoming Celtic chief executive Dominic McKay has taken part in the discussions ahead of his move from Scottish Rugby in July, with an appointment possible by next week.
Interim boss John Kennedy is currently preparing for Saturday night's Scottish Cup third-round tie against League One side Falkirk at Celtic Park.
The Parkhead club are set to undergo some radical changes at the end of the season.
Skipper Scott Brown will leave for a player-coach role with Aberdeen after signing a pre-contract agreement with the Dons.
Chief executive Peter Lawwell will also step down at the end of June with Dominic McKay from the Scottish Rugby Union taking over.
Celtic also announced on Wednesday night that Nick Hammond, their head of football operations, had resigned to "pursue new opportunities".