Solskjaer, who made no signings in January, has indicated he wants to oversee a clear-out at Old Trafford.
Ander Herrera and Antonio Valencia are leaving while Juan Mata is out of contract and his future at the club is still unresolved.
United ended the season poorly, with no wins in its final six games in all competitions, losing 2-0 at home to Cardiff City to conclude a Premier League campaign in which it finished sixth.
The Red Devils will not play in next season's Champions League, which is likely to deal further damage to any possible transfer budget, but Paul Pogba has been linked with a move away that could bring in a significant sum.
Woodward has indicated there will be funds for Solskjaer to invest in his squad regardless of player sales, given the club's continued commercial success.
"The strength of our business means we have the financial resources to continue to provide backing for the manager and creating success on the pitch," Woodward said. "This, as ever, remains our number one goal.
"It [the season] clearly didn't end the way we hoped, finishing in sixth place and with a disruptive managerial change partway through. However, Ole and the squad battled back from December to be in contention to qualify for the Champions League next season but, ultimately, we came up short.
"While the last few weeks were disappointing, we are delighted to have confirmed the appointment of Ole as our manager on a three-year contract, and to have confirmed key members of the coaching team, Mick Phelan, Michael Carrick, Kieran McKenna and Mark Dempsey will all be remaining at the club.
"Everyone at the club – the board, the manager, the squad and all the staff – are resolute in our desire to get United back to the top of English football. We continually look to improve staff on and off the pitch to achieve this."
Plans to revamp the Champions League have been met with widespread condemnation, with the Premier League, German Football League (DFL), French Football League (LFP) and European Leagues all criticising proposals to expand the group stage.
But Woodward indicated United, which will be playing in the Europa League next season, is not necessarily against changes being suggested by the European Club Association (ECA), which is led by Juventus chairman Andrea Agnelli.
"This is partly driven by domestic leagues across Europe not necessarily thriving and there being a desire from the clubs that are toward the top of those leagues to play more European games, which perhaps are more competitive," he said.
"I think there is drive from UEFA, which is laudable, which is to try to give greater access to more teams, so if you look at the year just about to finish, 80 teams competed in the two competitions [Champions League and Europa League] and the proposal for 2024 is 128 teams, so a greater than 50 per cent increase in the number of teams playing in Europe.
"There are other factors as well. From our perspective, we view these proposals as interesting but there's a lot of work to do with stakeholders to assess this. We're doing that ourselves together with our colleagues in the Premier League and the ECA."