Neil's role in doubt as Canaries take the drop.
Neil cut a disconsolate figure after Norwich's demotion was confirmed on Wednesday night despite a 4-2 defeat of Watford at Carrow Road.
The 34-year-old Scot was appointed in January 2015 and guided the Canaries to the top flight via the playoffs 12 months ago. He was noncommittal over his future in the immediate aftermath of relegation.
Norwich issued a statement saying their "immediate priority" was appointing a successor to David McNally, who resigned as chief executive following last Saturday's loss to Manchester United.
And the statement did not address Neil's future.
"The Norwich City board of directors and everybody at the club shares the pain and disappointment of our supporters," the statement read.
"We are proud of our team's battling and spirited display in the 4-2 win over Watford in our final home game of the season.
"We were immensely proud and humbled as well by the passionate support our manager Alex Neil, his staff and the players received from the fans, throughout the game and after the final whistle.
"However, as a club, our number one objective this season was always to retain our Premier League status.
"We have fallen short of that target and work is already under way to learn from the mistakes that contributed to our failure this season. The immediate priority, as stated earlier in the week, is to recruit a new permanent chief executive to lead the club."
Neil will sit down with Norwich owners Delia Smith and Michael Wynn Jones at the end of the season.
The win over Watford was just their third in 17 games, but Sunderland's defeat of Everton meant the Canaries are four points adrift with one game to play, at Everton on Sunday. Neil conceded Norwich were unable to strengthen the squad as much as he would have liked after ascending to the top flight via the play-offs, but pointed to individual errors and difficulty scoring as further reasons for their demise.
Norwich hope to respond to their relegation by returning to the Premier League at the first attempt. "The board and the whole club is fully focused on supporting and resourcing team-building plans for the summer transfer window and into next season," the statement added.
"The Championship is a very tough league as we all know, but our firm objective will be to bounce back and achieve promotion to the Premier League at the first attempt, as we did in 2014-15."
Captain Russell Martin, speaking following the Watford win, said on Norwich City TV: "The lads are all hurting. It affects the players and the staff at the football club because whatever it is that motivates you being in the Premier League is the pinnacle and now we're not there.
"Everyone is responsible and we need to learn from it. If you look elsewhere to try and deflect blame then you're kidding yourself.
"We've not been good enough over the course of the season. We have to move forward now because we've got a lot of good people at this football club.
"We have to learn from this season in order to progress. It's all about gearing up for next year and getting over this quickly."
Everyone is responsible - we need to learn from it. If you look elsewhere to try and deflect blame then you're kidding yourself Russell Martin
Steven Naismith of Norwich City looks dejected at Carrow Road on Wednesday night, after his side were relegated from the Premier League