The 22-year-old star has been on the radar of Portuguese manager Jose Mourinho since January when he was also being eyed up by EPL giants Chelsea.
And the Red Devils could still make their move one year later, after a disappointing summer transfer window.
Reports from the football news Italian website Calciomercato in Italy suggest AS Roma bosses have offered their key midfielder a significant pay rise in order to persuade him to remain at the club.
The Italian star has also been told to either remove or double the £26 million (€30m) release clause in his contract.
Inter Milan and Juventus are thought have joined Manchester United in the hunt for Lorenzo, but AS Roma are keen to keep hold of their star having already let go of central-midfielders Kevin Strootman and Radja Nainggolan earlier this window.
Jose Mourinho’s only major signing of Brazilain Fred has struggled so far at the Red Devils as a result of his £52 million move.