Rio Ferdinand feels Jose Mourinho had taken Manchester United as far as he could, while Gary Neville believes Mauricio Pochettino is the "outstanding candidate" to replace him at Old Trafford.
Mourinho was sacked by United on Tuesday morning, two days on from the 3-1 Premier League defeat at Liverpool, which left them 19 points behind the table-topping Reds in sixth place and 11 points off the top four.
Their current haul of 26 points from 17 games is their lowest ever at this stage in a Premier League season.
It is understood that while first-team coach Michael Carrick took training on Tuesday, United plan to name a caretaker manager from outside the club by the end of the week to see out the season, with a permanent appointment to follow.
Tottenham manager Pochettino and former Real Madrid boss Zinedine Zidane are among the names being linked with the Old Trafford hotseat.
Press Association Sport understands United are considering their former striker Ole Gunnar Solskjaer, currently manager of Molde, with regard to the caretaker role, and Pochettino is believed to be the main target to take
the reins next summer.
Former United captain Neville said on Sky Sports: "When I look at the values of Manchester United, you look at Mauricio Pochettino's belief in young players at (previous club) Southampton and with Tottenham. You look at his performance levels and the style of play, the way in which he carries himself at all times.
"People suggest he hasn't won a trophy yet at Tottenham but with a net spend of minus £29million or something over the last four years, I'm not sure he could have done more.
"I do think he is the person who is the outstanding candidate."
Pochettino himself on Tuesday insisted he is focused only on his job at Tottenham.
The Argentinian – under whom Spurs have finished third, second, and then third again in the last three seasons – told a press conference: "First of all I want to send my best wishes to him (Mourinho) and I feel so sorry because you know very well I have a very good relationship with him.
"Then it's not my business, what happened at another club. I only want to send my best wishes to Jose.
"We know a lot of rumours happen. But it's not my business what happens in another club now.
"I'm so focused on delivering my best job, our best job, in this club."
Liverpool boss Jurgen Klopp said at a press conference: "I can imagine the last few months were not a joy (for Mourinho). It is not nice to face all those questions.
"But nobody can take away all the things he has won. I hope he has that on his mind as he leaves. He is an outstanding manager."
Mourinho's first season with United saw them win the Europa League and League Cup, and they were Premier League runners-up last term.
Over the summer, the Portuguese was left frustrated by the failure to recruit the centre-back he craved.
Ex-Liverpool defender Jamie Carragher said on Sky Sports News: "I think you have to go back to the start of the season, and I just thought it was inevitable Jose Mourinho wouldn't be the manager the following season.
"There's been a lot of problems between him and the board and then his big-name players within the squad, there was a massive downturn in the relationship that was there for all to see.
"Taking the captaincy off Pogba, giving it to him in the first place, it just looked like it was coming to an end from the very start of the season, and that was before a ball was even kicked really."
Neville said of Zidane, who left Real over the summer having guided them to the Champions League title in each of the last three seasons: "He wouldn't be my choice in the sense I would choose Mauricio Pochettino, but on the other hand, you couldn't argue with the introduction of Zinedine Zidane because of his record."Neville also responded to Paul Pogba's deleted social media posts which came minutes after the news about Mourinho. United midfielder Pogba had posted a photo of himself to Instagram and Twitter looking at the camera with the words: "Caption this!", before swiftly deleting it. That prompted Neville to tweet: "Caption This" You do one as well!" There has been plenty of talk this season about the relationship between Mourinho and Pogba, who was stripped of the vice-captaincy in September and left as an unused substitute on Sunday. Former United striker Michael Owen wrote on Twitter: "Inevitable is the word I'd use. Mourinho had to go in the end. Simply not getting anywhere near enough out of his players."