Ten Hag was dismissed this morning after two-and-a-half years in charge with the 2-1 defeat by West Ham condemning United to 14th place in the Premier League.
United have won just three of their opening nine top-flight matches and are 21st in the new-look Europa League, having drawn their three opening fixtures.
Ruud van Nistelrooy, who returned to Old Trafford as part of Ten Hag's refreshed backroom staff in the summer, has been placed in interim charge.
However, the Manchester Evening News have claimed club figures are concerned they may struggle to buy a coach out of their contract.
Ten Hag could cost United around £17million in compensation and this may lead to a move for Solskjaer, who has been out of management since being sacked by the Red Devils in November 2021. Indeed, he is said to be in contention for a comeback.
But it is not clear whether the Norwegian, another club legend, is viewed as a serious long-term option as head coach or a backroom staff member.
Super sub Solskjaer spent nine seasons with United as a player, winning six Premier League titles and scoring over 100 goals to cement himself into Old Trafford folklore.
The 51-year-old took on the manager's role at the Theatre of Dreams in 2018 after a successful spell in interim charge following Jose Mourinho's sacking.
But despite guiding United to the Europa League final in 2020, he was sacked by the club in November 2021 after a 4-0 defeat at Watford.
Solskjaer recently revealed he would be open to returning to United.
Asked about taking the job by Norwegian outlet NRK in September, Solskjaer said: 'If your family needs you, then of course they will help.
'I would say "yes" any day of the week. It feels wrong to sit and talk about jobs that other people have. But of course, yes.'
Ruud van Nistelrooy (left), Ten Hag's assistant, has been placed in interim charge
United released an 87-word statement confirming Ten Hag's departure on Monday.
It read: 'Erik ten Hag has left his role as Manchester United men's first-team manager.
'Erik was appointed in April 2022 and led the club to two domestic trophies, winning the Carabao Cup in 2023 and the FA Cup in 2024.
'We are grateful to Erik for everything he has done during his time with us and wish him well for the future.
'Ruud van Nistelrooy will take charge of the team as interim head coach, supported by the current coaching team, whilst a permanent head coach is recruited.'