Michael Appleton has been in caretaker charge since Shakespeare's sacking last week and has overseen two wins in four days, with the Foxes beating Swansea City 2-1 on Saturday (21 October) and securing a spot in the quarter finals of the Carabao Cup after winning 3-1 against Leeds United.
However, according to the Telegraph Puel is set to become the club's third manager in this calendar year after Claudio Ranieri and Shakespeare, following talks with the Leicester board.
The ex-Saints boss has reportedly met with Leicester last week and on Monday (23 October) and could be officially appointed within the next 24 hours. The 56-year-old was sacked in June after just one season at St Mary's, despite having signed a three-year deal and leading the club to their first cup final in over a decade in the EFL Cup, which they subsequently lost 3-2 to Manchester United.
Puel also led Southampton to an eighth-place finish in the Premier League, albeit by racking up 17 fewer points his predecessor Ronald Koeman did in the previous season. The Frenchman, however, was criticised for what some perceived to be an overly defensive approach, which saw Southampton score just 41 goals in the Premier League, the worst attacking record of any side in the top 12.
Despite reservations over his style of play, however, Puel appears to have impressed the Leicester hierarchy and could be in the dugout when the Foxes welcome Everton at the King Power Stadium this coming Sunday (29 October).
Both clubs have endured a difficult season so far and Leicester will go into Sunday's clash just a point above Everton, who slipped into the relegation zone after losing 5-2 at home against Arsenal last week and which resulted in Koeman being sacked.
Meanwhile, Appleton is hopeful of remaining part of Puel's staff after guiding his side to two consecutive wins.
"It's been a rollercoaster and a difficult week and it would have been worse if we'd not won the two games," he told reporters. "I've been busy getting the players prepared and I'm sure there will be more information in the next few days. I'm not thinking too far ahead and we will be preparing for Everton on Sunday."