The Red Devils were top of the Premier League table in January but have since been overpassed by an unstoppable Manchester City.
Solskjaer's side are still alive in the FA Cup and Europa League, and the Norwegian still has support from the United board.
Speaking in a statement released alongside the latest set of financial results, Woodward said: "While the disruption to our operations remains significant, we are pleased by the tremendous resilience the club has demonstrated through the pandemic, underpinned by the dedication of our people and the strength of our commercial business.
"We have been reminded of the importance of football as a source of community, entertainment and pride to fans around the world, even as we have sorely missed them at Old Trafford. The progress made by Ole and the players this season is clear and our thriving academy and women's team are also adding to the optimism we feel about the future on and off the pitch."