The 20-time English champions are in the midst of their worst start to a season in 30 years, with a surprise 1-0 defeat last time out to Newcastle United leaving them just two points above the relegation zone.
That disappointing run has built up speculation on Solskjaer's future ahead of Liverpool's visit to Old Trafford, yet poor results and displays have not been enough to put the former Molde manager's job in immediate jeopardy.
Injuries have exposed a lack of squad depth at United but Woodward has come out and underlined his belief the club are on the right track in a recent address to staff at Old Trafford.
"The middle section of last season, after Ole's arrival, feels most relevant to what we want to achieve and where we want to be," he said.
"We saw a team playing fast, fluid football, with a clear representation of the style and philosophy the manager wants. Ole has also instilled the discipline back into an environment where we may have lacked it in recent years.
"He is building a squad that respects the club's history, in which players work hard and respect their team-mates. No-one is bigger than the club."
Recruitment has been the key factor behind United's floundering since Sir Alex Ferguson retired back in 2013, but there is now a belief behind the scenes that they are in a better place to target talent after particular issues during the reigns of previous managers David Moyes and Louis Van Gaal.
Speculation has continued over the potential appointment of a technical director as the club look at streamlining the top of the football department, but Woodward has hit back at those questioning the current structure and his involvement in deals at Old Trafford.
"There is a myth that we have non-football people making football decisions, and I think it's insulting to the brilliant people who work on the football side in this club," he added.
United will be hoping Solskjaer can ease the pressure with a positive result against Jurgen Klopp's Liverpool on Sunday.