"It is very hard to talk to the fans at this moment as they are hurting like us", Carvalhal said.
Carlos Carvalhal will reportedly leave Swansea if he does not keep the team up.
They have already been linked with a number of players as they aim to prepare their squad for the Premier League, but are there any players that play for the relegated clubs that they might look at?
Stoke sources have told i that the club's board remain undecided whether Lambert, the former Norwich and Aston Villa manager, is the right person to lead them in the Championship next season.
Swansea's owners have just released a statement following confirmation of the club's relegation from the Premier League earlier on Sunday afternoon. There have certainly been mistakes along the way and it is down to the owners to shoulder much of the responsibility. We believe this is a time for everybody to pull together even through the heartbreak and bitter disappointment of relegation.
"But we are still breathing a little, so let's see".
But while a late Gabriel Jesus goal downed the Saints, the Swans were unable to seal the three points that would have completed their great escape.
"There were 13 points before, with us we achieve 20".
"Once the surprise disappeared they played at the top, they were better than us".
Swansea will need to rebuild on and off the field next season, with the team no longer playing with the distinctive passing style once known as the "Swansea Way" and fan unrest representing the wider discord at the club.
"The players must show everybody they are Premier League footballers, not Championship footballers".
The 52-year-old will now take a few days out to reflect on the season before deciding whether he wants to have another go next term.
The home side's supporters chanted against absent chairman Huw Jenkins and American owners Jason Levien and Steve Kaplan all game and while a bright start at least promised a degree of hope, Swansea were made to pay for not taking their chances after Wayne Routledge missed a sitter.