After beating Hull in their first four Premier League contests, Sunderland has won just one of their last five against them, though this was the reverse fixture 3-0 this season.
Everton are understood to be preparing a £17m offer for Pickford, although all of the top sides are interested. Next week, Hull visits Palace and Swansea is away to Sunderland.
Several out of contract players can expect to depart, while goalkeeper Jordan Pickford and top-scorer Jermain Defoe, who both harbour World Cup ambitions, will look for top-flight suitors.
He said: "Good players and English players have a premium".
Moyes remains confident Pickford will not leave - warning potential suitors that would have "to pay the price".
"Obviously, Ellis was really disappointed [at relegation] because, whatever anybody says, he is really passionate about this club", he added.
"We can go into next week feeling much better and try to enjoy it". "We've got Marco Silva", they sang, though his future is just as muddied as that of Moyes.
"It's one of my proudest moments", said Clement, a former assistant at Real Madrid, Bayern Munich, Paris Saint-Germain, and Chelsea who is starting to look the part as a coach in his own right.
Player ratingsHull: Eldin Jakupovic: 6, Ahmed Elmohamady: 6, Andrea Ranocchia: 6, Harry Maguire: 7, Andrew Robertson: 7, Alfred N'Diaye: 5, Sam Clucas: 7, Lazar Markovic: 6, Kamil Grosicki: 7, Abel Hernandez: 6, Oumar Niasse: 5.
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"The guys show commitment again, but we played with pressure, and the other team didn't".
"I know what needs to be done to get back in the Premier League and I know what the requirements are to do that". Everyone talked about him and what a great prospect he was.
"If you're not calm, it is impossible to play well". That's impressed me as well.
"It gave us a lift before the game knowing that (Hull) result had gone in our favour; we knew if we did something special we could get out of the relegation zone". 'I think there is one more game to go. But our confidence is up and if we continue in this way I think we will stay in the league.
A number of impressive performances has led Pickford to be linked with a plethora of clubs, yet Everton are the first to really cement their interest in the player.
Pickford was not the only academy graduate on show at Hull.
David Moyes has confirmed that he will remain in charge of Sunderland next season.
"What we'd spoken about all week is the word "pride" for the team-mates and the club", said full-back Billy Jones after leaving the pitch. "We're looking at next season now, to take some good momentum into next season into next year, which is going to be a massive year for us".
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