According to Duncan Castles of the Daily Record, Mourinho is unhappy with "unnecessarily bureaucratic and inefficient organisation" at Old Trafford, as well as "hesitancy over implementing recruitment decisions and an unwillingness to match the funding" of United's rivals.
Mourinho is said to believe the Red Devils are not doing "everything realistically possible to improve performance on the pitch," while Woodward has yet to discuss an extension to Mourinho's three-year deal at Old Trafford, which expires at the end of next season.
A "close friend" of the coach's said talk of a new deal is "totally untrue, zero contacts for that."
The Portuguese told TF1 recently that: "What I can say is that I am still a manager with questions, with ambitions, with a desire to do new things and I do not think—I am sure—that I will not finish my career here."
Given Mourinho is only 54, it was always going to be unlikely he would stay at Old Trafford for the remainder of his career in the manner Sir Alex Ferguson did, with long-term managerial reigns like those of the Scot and Arsene Wenger at Arsenal a dying breed.
The Red Devils had a mixed first season under the former Chelsea and Real Madrid boss, as they finished sixth in the Premier League. But the campaign was largely salvaged by winning the UEFA Europa League, while the EFL Cup triumph provided a healthy boost too.
The club followed that up with a second summer of spending under Mourinho, and as Sporting Intelligence's Nick Harris revealed, Manchester City were the only domestic rival with a higher net spend:
United recruited well and have enjoyed a strong start as a result, going unbeaten in their first eight Premier League matches and consequently finding themselves just two points behind City. It's still too early for there to be concern that Mourinho has not been offered a new deal, though, which will likely come in due course if the team continues to prosper.
If Mourinho has serious complaints over the running of the club, it seems likely United will take note and make an effort to improve in that regard.
On the pitch, the responsibility lies with the manager, and as good as his players have been, he could be eking out a bit more.
He went to Anfield on Saturday with the goal of claiming a point, per football writer Tom Williams:
Oliver Holt of the Mail on Sunday was also disappointed, but BT Sport's Ian Darke saw merit in the United boss' approach:
Mourinho's caution in away games against rivals has typically brought him success, so it was of little surprise he adopted a similar mentality on Saturday, but Liverpool were there for the taking, and United will need every point they can muster if they are to beat City to the title.
The season may well be considered a failure if United don't win the league, and that could have ramifications for Mourinho's future at the club, but the manager is unlikely to walk away from what is arguably the biggest job of his career.
Mourinho may be unhappy with Woodward in some respects, but it would surely take a major breakdown in their relationship for him to consider leaving.