Director of football Jason Wilcox had initial, informal conversations with the former players on Tuesday - both of whom, it's thought, are keen on being involved again with the first team.
United do have other candidates they are considering for the role until the end of the season, including Ruud Van Nistelrooy who bridged the gap when Erik Ten Haag was sacked in October 2024 before Ruben Amorim took charge the following month.
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United look set to confirm a short-term appointment until the end of the season, to give them the breathing space to focus on a substantive appointment in the summer when more elite coaches are likely to be more readily available.
Oliver Glasner, Marco Silva and Andoni Iraola all see their current contracts expire at the end of the season, and Thomas Tuchel, Julian Nagelsmann and Mauricio Pochettino could be available once their international commitments are complete at the World Cup.
No meetings have been planned with Carrick and Solskjaer for Wednesday, with all the focus on Darren Fletcher's first game in charge of the team at Burnley. But it's expected those discussions will accelerate later in the week.
Solskjaer the favourite of interim options
Sky Sports News' football correspondent Rob Dorsett:
"Ole Gunnar Solskjaer would be a huge story if he came back. He was sacked by Manchester United in November 2021. He's incredibly popular with the Manchester United faithful.
"He loved it and he gave a very emotional farewell when he left. There were no bridges burnt. The impression I'm getting is that Solskjaer is the favourite.
"Jason Wilcox, who is leading this search, has spoken to him. Solskjaer has made it clear he would like to return to Old Trafford."
Glasner, Howe speak on United links
As talks continue for an interim boss, among those linked with the permanent role are Crystal Palace's Oliver Glasner and Newcastle's Eddie Howe.
Glasner has six months left on his deal at Selhurst Park but shut down any talk of the position during a press conference ahead of Palace's game against Aston Villa on Wednesday, live on Sky Sports.
"I am Crystal Palace manager. It's wasted time for you to ask any more questions. I don't talk about any other club other than Crystal Palace," he said.
Howe, meanwhile, also reiterated his commitment to Newcastle.
"I'm 100 per cent here, working as hard as I can, head down, ignoring all the stuff that you guys write, good or bad, because it's irrelevant," he said.
"It's about today's work and trying to prepare the team as well as we can for Leeds [on Wednesday], so that will never change from my perspective, my commitment to my work and my job.
"As long as I'm happy and able to express myself in the best way that I can to help the team, nothing will change."
Asked if there was anything which could tempt him away from St James' Park, Howe replied: "No, not at this current time.
"The most important thing for me is happiness in the role, happiness in the job, the relationships I have with the people around me. Now that's not always been consistently good and things can change at any football club.
"But, at the moment, I am very happy."
Fletcher: An honour to lead Man Utd
Current interim boss Fletcher gave his first news conference on Tuesday ahead of the midweek trip to Burnley.
After a busy 24 hours, the Manchester United U18s coach said: "It's been surreal. It's an amazing honour to lead a Manchester United team. It wasn't in my wildest dreams that it was something that could potentially happen.
"To lead out the team is an amazing honour and something I'm proud to do. It's not in the circumstances I expected it to happen so that doesn't sit easy with me.
"I've got a job to do which is to lead the team tomorrow. I have great honour and pride in doing that."
Asked whether he had any conversations with Wilcox and Omar Berrada, Fletcher replied: "I've had the conversations and the focus is on this game, then we'll speak after [Burnley].
"It all happened very quickly on Monday morning. Had a game to prepare and a very quick turnaround, a training session to organise, speak to the players so the focus has been on the Burnley match.
"It's important we give the Burnley match the respect and focus it needs to prepare the team in such a short space of time."
Fletcher also has the strange circumstance of having his two sons - Tyler and Jack - training with the first-team squad. Midfielder Jack already made his first-team debut under Amorim.
When asked if they could be handed full debuts, Fletcher started with a joke: "I won't be disclosing team information before the game!"
He then added: "They were in the training group on Monday because that was already arranged before I got there. They've been in the squad recently but we've got a couple of players back from injury, so we'll assess that tomorrow because of the short turnaround from games.
"They're in the periphery squad but whether they make the matchday squad, I don't know."
Man Utd's next five fixturesWednesday: Burnley (A) - Premier League, kick-off 8.15pm, live on Sky SportsSunday: Brighton (H) - FA Cup third round, kick-off 4.30pmJanuary 17: Man City (H) - Premier League, kick-off 12.30pm, live on Sky SportsJanuary 25: Arsenal (A) - Premier League, kick-off 4.30pm, live on Sky SportsFebruary 1: Fulham (H) - Premier League, kick-off 2pm, live on Sky Sports