The Premier League champions received an unconditional offer of $131 million for Ronaldo, and unlike last offseason is willing to see its 24-year-old star join a major European rival.
United said: "At Cristiano's request - who has again expressed his desire to leave - and after discussion with the player's representatives, United have agreed to give Real Madrid permission to talk to the player.
"Matters are expected to be concluded by 30 June."
Madrid confirmed the offer for Ronaldo in a statement, saying: "The club hopes to reach an agreement with the player in the coming days."
The Portugal winger, who joined United from Sporting Lisbon in 2003, spent last June pushing for what he called at the time a "dream move" to Madrid.
An unsuccessful complaint about Madrid's pursuit was made to soccer's world governing body FIFA and a move was blocked by United manager Alex Ferguson, who traveled to Lisbon to persuade his player to stay at Old Trafford.
That didn't stop Ronaldo being linked with Madrid, and Ferguson's irritation boiled over in December when he said he "wouldn't sell Madrid a virus," describing the club as a "mob."
But now a world-record bid appears too hard to resist for a club whose debts have spiraled to nearly $1 billion.
Ronaldo, who was under contract until 2012, plunged his future into doubt after United lost the Champions League final to Barcelona last month, saying he wasn't sure if would stay. This came despite having vowed to remain with United in the buildup to the match and even in a broadcast interview conducted with himself.
The return of Florentino Perez as Madrid's president has changed things as the billionaire looks to spend lavishly to return the Spanish side to the glory days of its "galactico" era.
Brazilian star Kaka was signed away from AC Milan earlier this week for a reported fee of around $92 million.
But the offer for Ronaldo would eclipse that, as well as the $65 million Madrid paid to lure former France striker Zinedine Zidane away from Juventus in 2001.
Ronaldo's arrival could strengthen Madrid's position as world soccer's richest club based on revenue. According to accountancy firm Deloitte, Madrid earned over $512 million compared to second-place United's $455 million.
On the pitch, though, Ronaldo would be joining a club which just had its first trophyless season in three years, while United won a third straight Premier League title, the Club World Cup, and the League Cup.
Ronaldo finished as United's top scorer again, despite not matching the 42-goal haul in the 2007-8 campaign when Ferguson's side won the Champions League and the English title.
However, Ronaldo was viewed by some as a divisive influence at United, openly challenging Ferguson's authority.
In the last weeks of the season he angrily swiped at some TV equipment while shaking his head in clear annoyance when substituted against Manchester City. When he missed a shot against Wigan, he answered Ferguson back from the pitch while shrugging his shoulders.
Former United manager Tommy Docherty said Thursday that relations were clearly strained with Ferguson, as they had been with David Beckham before he was allowed to join Madrid in 2003.
"You can't have disharmony," said Docherty, who was manager between 1972-77. "With ($151 million), they should cut their hands off with Ronaldo and run and give Real Madrid the problem."