The Portuguese coach is now expected to talk to the Foxes after parting company with the Swans "by mutual consent", though it is unclear whether the Welsh club will be entitled to compensation.
"The move comes following a meeting between Swans chairman Huw Jenkins and Sousa," read a statement on Swansea's official website.
"Both parties agreed that the current speculation regarding the manager's future was harmful to the club and, as a result, Paulo was granted his request to speak to Leicester"
Coaches Colin Pascoe and Alan Curtis will take charge of the first-team in a caretaker basis.
Middlesbrough have confirmed the signing of forward Kris Boyd from the Scottish Premier League side Glasgow Rangers.
Boyd, 26, was out of contract at Rangers and has signed a two-year contract at Boro.
Hull City have parted company with Geovanni as they continue their preparations for life in the Championship.
The Brazilian, 30, joined the Tigers from Manchester City two years ago and went on to score 13 goals in 65 appearances for the club, including their first goal in the Premier League - against Fulham in August 2008.
It is understood that Geovanni is set to sign for a Turkish side after agreeing to bring his £30,000-a-week KC Stadium contract to an end.
Norwich City have completed the signing of goalkeeper John Ruddy from Everton for an undisclosed fee. Ruddy, 23, has signed a three-year contract at Carrow Lane.
Nottingham Forest have accepted a bid from Newcastle United for the defender James Perch.
Gethin Jenkins, the chief executive of Cardiff City, has said the club are on course to settle £1.3million of debts to Her Majesty's Revenue and Customs.
The Bluebirds are due in the High Court on August 11 to face another winding-up order over the amount - which stems from not paying their share of income tax.
The former Cardiff forward Josh Magennis has joined Aberdeen. Magennis, 19, had made his Bluebirds debut in the 4-0 win over Scunthorpe on the opening day of last season.