According to the report, newly-appointed Sunderland manager Simon Grayson is a fan of the 23-year-old midfielder, who is claimed to be rated at £2 million.
Woods is a good midfielder who can also operate as a defender, and he would be a smart signing for the Black Cats.
Sunderland will feature in the Championship next season after getting relegated from the Premier League at the end of the 2016-17 campaign.
The former Shrewsbury Town man has good experience in the Championship with Brentford, and Sunderland need players like him in order to finish in the top two and gain automatic promotion to the Premier League.
If Woods is available for £2m, then it would be wise of Sunderland to make a move and snap up the midfielder in the summer transfer window.
For the Black Cats to get out of the Championship next season, they need players who have extensive knowledge of the division.
Meanwhile, Sunderland would have to splash out as much as £3 million in order to sign Max Gradel, who is 29 years of age and can operate as a winger or as a forward, if they are unable to reach a loan agreement with Premier League club Bournemouth, according to The Shields Gazette.