Manchester United legend Ryan Giggs is eager for talks with Southampton.
The Daily Mail says Giggs would relish the chance to take his first steps into full-time management on the south coast, given the club's strong infrastructure.
Former Borussia Dortmund boss Thomas Tuchel was initial favourite to succeed Puel but he wants to wait for a Champions League club.
Frank de Boer and former Alaves boss Mauricio Pellegrino are also on the shortlist.
Puel was officially sacked on Wednesday following prolonged speculation but it is understood that there was a feeling within the club that he had been told last week his spell at St Mary's was over after one season.