Solskjaer admitted he was "embarrassed" after his team's fifth defeat in their last seven Premier League games.
The Times, The Guardian and the Manchester Evening News reported that the result of an emergency board meeting called after the Watford defeat was to bring the Norwegian's three-year spell in charge at Old Trafford to an end.
Solskjaer signed a new three-year contract in July and will reportedly be handed a Â£7.5 million compensation package.
The club's hierarchy stuck by Solskjaer during the recent international break despite humiliating home defeats to Liverpool and Manchester City.
But worse was to come at Vicarage Road as the struggling Hornets could afford to miss a first half penalty twice and still run out comfortable winners.
Solskjaer has managed to ride out the storm during previous poor runs since taking charge in 2018.
But United's latest slide comes after heavy pre-season investment on Cristiano Ronaldo, Jadon Sancho and Raphael Varane that were expected to turn the team into title-winners for the first time since 2013.
Financial results released this week for the first quarter of the season showed United's wage bill has risen 23 percent.
However, that has not been rewarded on the field as the Red Devils find themselves 12 points adrift of Premier League leaders Chelsea down in seventh and out of the League Cup.
Much more was expected after steady progress during Solskjaer's first two full seasons in charge.
He secured back-to-back top-four finishes in the Premier League for the first time since the club's decline began with Alex Ferguson's retirement as manager in 2013.
However, Solskjaer, who famously scored United's winning goal in the 1999 Champions League final against Bayern Munich, has failed to win a trophy since his return to the club.
His legendary status as a United player meant the 48-year-old retained the support of fans even as results went downhill in recent months.
But even they began to turn on him on Saturday as he was booed while holding his hands up in apology.
"We were outfought," said Solskjaer. "The first half is the worst we've played and it's hard for me to explain why we played the way we did.
"The boys are in a terrible place in their heads now -- we have let ourselves down and the fans down. It's hard to stand here and explain that, but that's football and we have to take the flak for it."
Former Real Madrid boss Zinedine Zidane, Leicester manager Brendan Rodgers and Ajax coach Erik ten Hag are the frontrunners to be Solskjaer's successor.
Standout matches today