Shearer failed to keep the Magpies in the Premier League during his eight-game reign after Sunday's 1-0 defeat at Aston Villa confirmed their drop to the Championship.
A downcast Shearer refused to discuss his future after the match and suggested he would think long and hard about whether to offer his services for next season.
Shearer said "big mistakes" had been made and that "major changes" were required to revitalise the club and managing director Derek Llambias has said that initial discussions about the future will begin this week.
"Being relegated from the Premier League is a huge disappointment for everyone involved with Newcastle United," Llambias told the club's official website.
"We are all hurting and I feel desperately sorry for everyone associated with Newcastle United; for [owner] Mike [Ashley], who has invested heavily in the club, and for all the supporters who have given the team magnificent backing up and down the country all season long.
"Mike and I will sit down with Alan this week to discuss how the club moves forward again and I hope to be able to say more to our supporters later this week."
The Middlesbrough manager Gareth Southgate said he will make major changes to the squad if given the chance to manage them in the Championship next season.Boro slipped out of the Premier League after losing 2-1 at West Ham United on Sunday.
Nothing has been discussed with the chairman, we have to sit down this week and plan more definitely what is going to happen."It will be a tough challenge and we'll have to improve. But we're ready to make those changes because it clearly hasn't worked this year," he said.
"There will be personal agendas and as a club we will have to bring some money in to go forward.
"There are also situations I want to change. Inevitably, there will be big changes to the playing squad."