The Welsh club are understood to be keen to appoint a manager with Premier League experience and former West Bromwich Albion manager Tony Pulis is also in the frame, as is Slaven Bilic, who was sacked by West Ham United last month.
Ronald Koeman, another free agent after being shown the door at Everton last month is also among the front runners, while Leon Britton, who remains a Swans player despite taking up a role on the coaching staff, is also an option.
Over the weekend, Van Gaal revealed that while he is unlikely to return to coaching, he would make an exception for a Premier League club, as it would give the chance to "get one over" his former club.
The 20-time champions of England sacked the former Bayern Munich manager in May 2016 soon after he had guided them to a first trophy in three seasons, as United clinched the FA Cup by beating Crystal Palace 2-1 at Wembley.
"I will probably not manage a club anymore," Van Gaal was quoted as saying. "I would make one exception — if a big English club comes for me, than I would do it. Because then I can get the chance to get one over on Manchester United.''
The Dutchman, who spent two seasons at Old Trafford, has not forgiven United's chief executive Ed Woodward for approaching Jose Mourinho behind his back and for the way the club handled his dismissal.
Four defeats in the last five league games have left Swansea bottom of the table four points from safety and Clement, who has won just three games this season, was relieved of his duties less than a year after taking charge at the Liberty Stadium.
"We had three different managers last season and as a result we all wanted to give Paul as much time as possible to turn things around," the club's chairman, Huw Jenkins, said in a statement on Wednesday.
"But we felt we couldn't leave it any longer and needed to make a change to give us the best chance of an uplift and a turnaround in fortunes with the club bottom of the Premier League.
"Paul has been at the club for 12 months and what he achieved in the second half of last season to keep us in the top flight was a tremendous feat. For that, and his effort and commitment this season, it goes without saying that the club thanks him for his work, together with Nigel (Gibbs) and Karl (Halabi).