Frank De Boer became the bookmakers' favourite yesterday and is prominent on the shortlist but the club have also spoken to Aitor Karanka, the former Middlesbrough manager, and will continue to assess their options over the next few days.
Swansea's owners will be banking on fresh impetus from a new boss to lift the team up the table, with the likes of Leicester and Everton boosted by new appointments in recent weeks.
"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".
Jenkins praised Clement's role in helping keep the club in the English top flight last season.
Clement was sacked by the Premier League's bottom placed club on Wednesday after just under a year in charge.
It would be an ambitious move for the Swans as Martinez is now head coach of Belgium and will be taking them to Russian Federation for next summer's World Cup finals.
Paul Clement cut a dejected figure as he watched his Swansea City squander a goal advantage to eventually lose 3-1 to a resurgent Everton at Goodison Park last night, and the Premier League basement boys, who have lost four of their last five top-flight matches, will need to quickly recapture some sort of positive form if they are to escape relegation.
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).
But the summer departures of Gylfi Sigurdsson, Fernando Llorente and Jack Cork significantly weakened Clement's hand, and the former Derby boss had hoped to be given the January transfer window to strengthen his squad.
The club said they would give supporters a further update within 24 hours.