Joshua King fired in an 88th minute winner to end their 10-year stay in the top flight.
Moyes is understood to be considering his future after a disappointing return to England in the north east.
David Moyes is set to quit Sunderland after suffering relegation
David Moyes has struggled after replacing Sam Allardyce at Sunderland
And The Sun now report that he will quit – and could be gone by the end of next week.
Moyes met with club chiefs yesterday and could leave the club in lurch for their remaining games.
Sunderland have four games to play and sit on 21 points – 13 points shy of fourth-bottom Hull.
Sunderland transfers: 16 players who could leave after relegation Sat, April 29, 2017
Express Sport runs through 16 players who could potentially leave Sunderland after their relegation from the Premier League
Play slideshow REUTERS 1 of 17
Who will leave Sunderland after their relegation from the Premier League?
Moyes replaced Sam Allardyce at the Stadium of Light when he joined England.
But, following disappointing spells at Real Sociedad and Manchester United, Moyes was unable to turn the corner.
Sunderland face a player exodus which will be led by Jermain Defoe and Jordan Pickford and Moyes fears he won’t be able to get them back up.
Aitor Karanka was given the boot by Middlesbrough in March but has been linked with Sunderland
Spanish newspaper Sport are claiming Sunderland are considering a shock move for ex-Middlesbrough boss Aitor Karanka.
Karanka was sacked by Boro in March after struggling in his first season in the top flight.