In just two days, their efforts have made a significant impact, benefiting over 10,000 people in flooded areas
Videos shared on social media on Tuesday show hooded people throwing flares over a gate to a property, while red paint was sprayed on a gatepost.
A statement from Cheshire Police said officers were notified of an incident of criminal damage at around 10:45 pm (2245 GMT) on Tuesday.
"Thankfully no one was harmed and officers will liaise with security officials over the coming days to establish the full circumstances surrounding this incident and identify those involved," the statement said.
Woodward, who is married with two young children, is reported to have been away from his home at the time of the attack.
United said those responsible would be banned for life from the club.
"We know that the football world will unite behind us as we work with the police to identify the perpetrators of this unwarranted attack," the club said.
"Anybody found guilty of a criminal offence, or found to be trespassing on this property, will be banned for life by the club and may face prosecution.
"Fans expressing opinion is one thing, criminal damage and intent to endanger life is another. There is simply no excuse for this."
The atmosphere at Old Trafford has turned toxic in recent weeks, with supporters voicing their anger at United's owners the Glazer family and Woodward, who advised the Americans during their successful takeover of the club in 2005.
Fans reportedly sang "Woodward's going to die" at the recent defeat against Burnley and during Sunday's FA Cup victory over Tranmere.
The supporters, who enjoyed two decades of almost unbroken success under Alex Ferguson, have seen their club fall 33 points behind fierce rivals Liverpool, who look certain to win their first league title for 30 years.
United manager Ole Gunnar Solskjaer is struggling to turn things around, with the club fifth in the Premier League, six points behind fourth-placed Chelsea.
Their record following former United striker's Solskjaer's appointment as permanent manager in March last year is poor, with United having lost 12 Premier League games - one more than they have won - under the Norwegian in that time.
United have not won the Premier League since Ferguson retired in 2013, with far more experienced managers David Moyes, Louis van Gaal and Jose Mourinho all in and out at Old Trafford before Solskjaer took charge.
They now face Manchester City in the League Cup semi-final on Wednesday but are 3-1 down after the first leg.
Earlier this month, Solskjaer gave his backing to the Glazers and Woodward after fans voiced disapproval at their handling of the club.
"As a club we've got to stick together, we've got to be united, we are a family," he said.
"I can only say from when I've been here I've been backed by the owners, I've been backed by Ed and they're supporting me, so for me, make sure they stick together."
In just two days, their efforts have made a significant impact, benefiting over 10,000 people in flooded areas
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