The 42-year old has swiftly cemented himself as favourite for the vacant role, with comments made following Real Madrid's Champions League win suggesting that he is ready to move into management.
The coach started his career at Chelsea, as an academy coach in 1996. Following Guus Hiddink's arrival at Stamford Bridge, the former teacher was appointed first-team coach - a position he held during Carlo Ancelotti's reign. Clement followed Ancelotti to PSG - becoming his assistant in the process - and then on to Madrid.
A successful season has seen Clement help Madrid to victory in the Copa del Rey as well as the UEFA Champions League.
However, the Englishman appears to wish to enter the realm of management, suggesting to Sky Sports that he wants to be a manager by 2017.
Clement said that Ancelotti has also been the catalyst for a desire to move into management.
He said: "Carlos said to me that I have to give it a go, which has given me confidence. If he thinks I can do it then why not?"
And following the extra-time victory over city rivals Atletico Madrid, Clement once again laid down his plans to become a full-time manager, as reported by The Argus.
Despite a glittering career as an assistant, with league titles in England and France - as well as the famed European Cup victory - it is unlikely a top division club would be willing to risk appointing Clement.
His lack of actual management experience makes him a better fit for Championship club Brighton, a place where he can further both his career, as well as the club, without the increased expectations that come from coaching in the Premier League.
For these reasons, The Argus has quoted Clement as shortening to odds of 5/4 this morning, moving ahead of Tony Mowbray, Neil Lennon and Tim Sherwood as successor to departed head coach Oscar Garcia.
Should Paul Clement leave Real Madrid for Brighton?