The Daily Mirror is reporting the Swans want the Dutchman to take over at the club cemented to the bottom of the Premier League.
The paper goes on to say the South Coast hierarchy have put the former Netherlands boss on their shortlist and are hoping to make an appointment before Saturday's game with Crystal Palace.
However, the Daily Star report Clement's sacking paves the way for Tony Pulis to return to the Premier League.
The media outlet states the manager has a reputation for keeping clubs safe from relegation and could make a swift return to management after being dismissed by West Bromwich Albion last month.
Antonio Conte is keen to bring in three new faces to Stamford Bridge with the paper reporting Bayer Leverkusen star Leon Bailey and Monaco's Thomas Lemar also subject to interest.
Elsewhere, Alexis Sanchez will be heading to Manchester City in a £400,000-a-week deal, the Daily Mirror says.
The Chilean striker - whose contract with Arsenal expires in the summer - will become one of the game's top earners in the deal which City can afford as they will not have to pay a transfer fee.
The England youth international has been on the books of Manchester United since 2010.