United are top of the Premier League table by three points over rivals Liverpool after 17 games.
Moves for the young Englishmen would cost United more than £100m in any January or summer transfer this year.
Solskjaer has been given a massive boost by the board, according to ESPN - but it comes with a warning.
Woodward has said to the boss that the club are “willing to spend" as they look to strengthen the squad for the 2021/22 season.
But the report states that staff in charge of budget will remain “cautious" due to the impact of the UK lockdown.