Media playback is not supported on this device
The Black Cats' 10-season spell in the top flight ended when they lost 1-0 at home to Bournemouth on Saturday and Hull City drew 0-0 at Southampton.
Moyes, who took charge at Sunderland in July last year, has faced calls from supporters to step down.
"I think it's a question for two or three weeks' time," said the Scot.
"I think we all need to reflect and look at everything from top to bottom."
Chairman Ellis Short, who has been trying to sell the club, apologised to Sunderland fans, and admitted mistakes had been made.
Since Short bought the club in 2009 - he became chairman in 2011 - the Black Cats have had seven full-time managers, but have faced the threat of relegation in almost every season.
"We need to improve both on and off the field, and despite the bitter disappointment there is a strong determination to do so throughout the club," said the American.
"There is significant work to be done over the summer and when the season is concluded, we intend to share our plans to move forward with our supporters."
Listen as fans react to Sunderland's relegation on 606
This is the first time Moyes has been relegated as a manager, and he warned fans just two games into the the season that he thought they would struggle.
"I feel for the supporters because they're the people who pay their hard-earned cash to come and watch and we've not given them enough this season," said the former Everton and Manchester United manager.
"I've had 400-odd games in the Premier League and I've got an idea of what a good squad looks like.
"My feeling at the start of the season was it was going to be a hard graft, and I'd rather be up front with people than tell them something different."
The defeat by Bournemouth was the ninth time Sunderland have failed to score in their past 10 games, and the 18th time they have failed to do so this season.
Jermain Defoe, the team's top scorer this season with 14, has not found the net in more than 15 hours of Premier League football.
"We've needed Jermain's goals," said Moyes.
"At times we haven't given him enough quality supply. But there has been other times, like today, when a couple of chances came. In the early part of the season he was getting them and finishing them."
Moyes was given the Sunderland job after Sam Allardyce  left for his brief stint as England manager.
He has not been helped by injuries, with Jan Kirchhoff, Lee Cattermole, Duncan Watmore, Jordan Pickford, Paddy McNair and Victor Anichebe among the first-team players to have had lengthy spells on the sidelines.
Short said: "I acknowledge that during my ownership mistakes have been made, particularly in the area of player recruitment, and as a result we have found ourselves struggling to survive in recent seasons.
"We had massive disruption during the summer transfer window, and an unprecedented number of injuries throughout the season.
"These are difficulties which we have been unable to overcome and we are paying the price for that now."
Media playback is not supported on this device
Match of the Day pundit and former Newcastle striker Alan Shearer:
David Moyes has got to take responsibility as has the owner. It's been a disastrous season from start to finish. Mismanagement at all levels and they've signed some poor players. They've been flirting with relegations for a number of seasons and it was inevitable it was going to happen.
A £28m net spend in the last three transfer windows is comparable with teams in and around them, but I worry for Sunderland. There are two players that are assets in that team [Jordan Pickford and Jermain Defoe]. Other than that there's not too much in that squad that's worth a lot of money.
Match of the Day commentator John Motson:
A club badly managed from above for the last four years. At least three managers have got them out of it at the end of the season.
The rot set in at Sunderland a long time ago and it came right from the top.
I have no sympathy with them - they deserved to do down. I just hope somebody, whether it's David Moyes - a new owner perhaps - has got the opportunity to bring them back.
They have been a great club, they've got great fans but I'm afraid the way the club has been run, they've deserved what they got.
Former Premier League striker Jason Roberts:
Those Sunderland fans are seeing a team and a club that I think has lacked leadership from the boardroom.
That's why they have skipped from manager to manager with no real plan or identity.
Maybe they need to go down, refocus, bring in some young hungry players and give a new "project" to the Premier League because Sunderland are a huge club and if they come back up next season, they will be a club many will fear.
Sunderland manager David Moyes said it was "too soon" for him to commit to the club for next season, following their relegation from the Premier League.