Clement has been without a club since leaving Derby County in February.
But the tabloid claims the former Real Madrid assistant coach is a now a target for Swansea and says he has already entered talks about taking over when Guidolin's contract expires.
Former Swansea manager Brendan Rodgers also features on the club's shortlist according to the Sun, alongside Sheffield Wednesday head coach Carlos Carvalhal.
The report also claims that Guidolin could even stay at the Liberty Stadium, with the Italian also holding discussions about his future after steering the Welsh side away from the Premier League's bottom three.
Clement, 43, has been working alongside Gareth Southgate for England Under-21s since the culmination of his first managerial role.
The highly-rated coach began his career at Chelsea's academy and worked under former Blues boss Carlo Ancelotti at Paris Saint-Germain and Madrid, where the pair lifted the Champions League trophy three years ago.