According to the report, Hodgson, who was sacked as England manager in July, held talks with representatives of the reigning Premier League champions in London last Friday.
Leicester City owner Vichai Srivaddhanaprabha is believed to be a big admirer of the 69-year-old, with the Thai Billionaire believing that his team needs experience following Leicester’s horrendous title defence.
The 70-year-old is keen to return to management following his England sacking after their embarrassing 2-1 loss to Iceland in July.
However, current Leicester caretaker manager Craig Shakespeare is expected to take charge of the Foxes home game against relegation rivals Hull City on the weekend.
Hodgson is the bookmakers favourite to succeed Claudio Ranieri, who was sacked less than a week ago following a run of bad defeats.
It is believed that talks will carry on in the coming weeks.