The report claims the two parties will hold informal talks over the next few days to discuss the possibility of a transfer.
Ronaldo recently downplayed reports that he's agreed a new Real Madrid contract, tweeting: "The news about my renewal with Real Madrid is false."
And despite repeated assurances from Real Madrid president Florentino Perez that Ronaldo will stay and finish his career in the Spanish capital, the Daily Express claims he's open to holding talks with United.
FIFA rules expressly forbid clubs from meeting directly with players who hold contracts with other teams, but the report suggests unofficial talks could begin within seven days.
United have to establish whether they possess the finance to complete such a deal, with the transfer fee involved likely to be a British record.
But there is another option. The Premier League champions may sound out Ronaldo's interest in moving back to his former club further down the line, with the aim being to wait until his Madrid contract expires.
That would free up Ronaldo to move back to Old Trafford - and in January 2015 he would be able to hold pre-contract talks over an agreement with another team.
United are not the only club interested in Ronaldo, and the Daily Express report claims Paris St Germain are ready to offer the Portugal international £225,000-a-week to leave Madrid.