City chairman Peter Ridsdale admitted in January that both Manchester United and Arsenal were tracking the highly-rated youngster, but that a deal had yet to be agreed with either.
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, and appear likely to repeat the performance of June 2008 when Arsenal managed to lure former Bluebirds star Aaron Ramsey to the Emirates Stadium ahead of the Red Devils.
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"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 is to strengthen the squad. We don’t want to let any key players go."
Swansea-born Matthews has made 27 league outings for Dave Jones' side in 2009/10 and featured for Wales U21s, helping him win the Championship Apprentice of the Year prize at Sunday's Football League Awards.
However, when asked about the high-profile clubs being linked with his signature, he ensured he kept his cards very close to his chest.
"There's been a few rumours but nothing has been cemented," he told host Mark Clemmit on stage upon receiving his award.
"Maybe something will happen in the summer, but I'm happy at Cardiff, so we'll see."
Matthews edged out Watford's Lee Hodson and Sheffield Wednesday's Nathan Modest to pick up the Championship Apprentice prize, with Norwich midfielder Tom Adeyemi and Cheltenham's Kyle Haynes taking the titles for Leagues One and Two respectively.
Players eligible for the prize are those in the second year of their apprentice (i.e. be in his 'Under 18' year and be registered as either a second year scholar or as a first year professional. Judges looked at the record of the player in all competitions since he became an apprentice alongside his educational attainment, disciplinary record and any other outstanding achievements.
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GALLERY:Photos from Football League Awards 2010
