A statement on United's official website said: "Manchester United is delighted to announce it has agreed terms with Cardiff City for the transfer of Aaron Ramsey subject to terms being agreed and the player passing a medical."
A statement from FA Cup finalists Cardiff on Tuesday said: "We can confirm that on 3 June the club has received formal bids from three top Premier League teams for Aaron Ramsey.
"And Aaron has been given permission to speak to two of these teams.
"One of these teams is Manchester United though at this stage we can confirm that no deal has been done and the player has not yet agreed terms."
Cardiff would prefer to sell Ramsey to the European champions because as part of their deal, Sir Alex Ferguson would loan the player known as "Rambo" back to The Bluebirds to gain vital first-team experience.
Ferguson has kept a close eye on the young prodigy since Ramsey, the most wanted man outside the Premier League, broke into the Cardiff first-team this season.
Cardiff boss Dave Jones and chairman Peter Ridsdale met Ferguson when Cardiff's management team travelled to Barcelona in April before United's Champions League semi-final.
Any Ramsey signing would be the second deal struck between the two clubs this summer when Cardiff signed United's reserve team goalkeeper Tom Heaton on a season-long loan.
Uncapped Ramsey was included in John Toshack's full Wales squad for last week's friendly double-header against Iceland and The Netherlands but did not feature.
Ramsey has played just 22 games for Championship side Cardiff but has been touted as Ryan Giggs' heir apparent in the Wales side and the Caerphilly-born Welsh speaker could follow his idol and fellow countryman at Old Trafford.
Speaking on behalf of the national set-up Wales assistant manager Dean Saunders said they would prefer any deal for Ramsey to include the option of him being loaned back to Cardiff for next season.
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