Both Sir Alex Ferguson and Arsene Wenger are believed to be interested in the £4 million-rated defender, who has broken onto the first-team scene in South Wales this season - making 17 league outings.
The pair appear likely to repeat the performance of 18 months ago when Arsenal managed to lure former Bluebirds star Aaron Ramsey to the Emirates Stadium ahead of the Red Devils.
And with United’s defence currently decimated through injury, 17-year-old Matthews could be used to help fill the void.
His impressive performances at right-back for the Championship side have helped Dave Jones' side rise to 4th place in the standings.
Swansea-born Matthews progressed through the Cardiff youth team before earning first team honours initially in April 2009.
He also made his debut for Wales U21s in a 2-1 defeat to Bosnia-Herzegovina last November.
However, Ridsdale refuted reports that either Manchester United or Arsenal have already agreed a deal with their prodigy, but confessed that there has been interest.
"We’ve had two or three enquiries and I know Manchester United and Arsenal are in for the lad," said Ridsdale.
"But there is no agreement with any club, verbal or otherwise. We want Adam to stay with Cardiff and that’s what we are hoping will happen.
"Our aim this month is to strengthen the squad. We don’t want to let any key players go."
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