Carvalhal and Lambert leave
Swansea chairman Huw Jenkins says it is "vital" the club appoint "the right person" as Carlos Carvalhal's managerial successor.
Swansea have announced that they will not be extending Carvalhal's contract.
The Swans were relegated from the Premier League on Sunday, the end of their seven-year stay in the top flight confirmed by a 2-1 home defeat to Stoke.
Carvalhal was appointed as Paul Clement's successor in December, just three days after leaving Sheffield Wednesday.
He won five of his first nine league games as Swansea climbed to 13th place, but they were condemned to the Sky Bet Championship after picking up only three points from the last 27 on offer.
Managers already being linked with the post include former Wales boss Chris Coleman, who was recently sacked by Sunderland, and Swedish club FK Ostersunds' boss Graham Potter, while long-serving Swans player Leon Britton also has his admirers.
Jenkins said: "While there are already rumours linking us with several managers, it is vital that we identify the right person to take this team forward."
Meanwhile, Paul Lambert has left his post as Stoke manager by mutual consent following their relegation to the Championship.
The Scot was appointed on a two-and-a-half-year deal in January but was unable to save the Potters from sliding into the second tier after a 10-year stay in the top flight.
Lambert had denied there was a break clause in his contract that could be activated by him or the club in the event of Stoke going down, but the Potters will now be looking for a new manager ahead of next term.
Former Aston Villa and Norwich boss Lambert won just two of his 15 games at the helm.
A club statement read: "Stoke City can confirm that Paul Lambert has left his position as manager by mutual consent."