The France international was linked with a potential exit at Old Trafford throughout the summer transfer window.
Reports in the British media suggested that Martial was keen to leave Manchester United due to a lack of first-team football.
Martial was touted as a potential target for Inter Milan, Juventus and Tottenham Hotspur but a deal failed to materialise.
The France striker has made one start and one substitute appearance in the Premier League in the current campaign.
Daily Record journalist Castles revealed that Woodward was one of the chief reasons that Martial stayed at Old Trafford.
“He was not sold because Woodward wanted to retain him at the club and feels he is a player who should be developed at Manchester United and not allowed to go elsewhere,” Castles told the Transfer Window podcast.
“The player wanted to leave, the agent was trying to get a move and the manager felt the most pragmatic thing to do would be to take the deal on the table – and there were a number of deals on the table.”
Manchester United signed Martial in a £36m deal from Ligue 1 side AS Monaco in the 2016 summer transfer window.
The French forward scored on his debut in a 3-1 win over Liverpool FC under Jose Mourinho’s predecessor Louis van Gaal in 2015.
The Red Devils forward has won the Europa League and the League Cup since his move to Old Trafford.
MORE: Man United latest news
MORE: Arsenal latest news
MORE: Chelsea latest news
MORE: Liverpool latest news