The Premier League side are said to have had a summer approach for the Argentine rejected by the Rome club, but are set to return once the transfer window re-opens.
The figure offered was in the region of £13m and it’s likely that they will up that sum to tempt the Serie A giants.
Despite Roma suffering an underwhelming start to the season, Lamela has bagged eight goals, a total that no other player in the division has been able to equal.
The 20-year-old is regarded as one of the top emerging talents in world football and has been previously compared to Barcelona star Lionel Messi.
He joined his current club in 2011 from River Plate for an estimated fee of £12m plus add-ons.
With Roma struggling to move forward, Lamela may be tempted to jump ship to increase his chances of making the Argentinian squad for the 2014 World Cup.
United boss Sir Alex Ferguson is said to see the tricky wide man as a replacement for Nani, who is expected to leaving the club in the not too distant future.
Roma are believed to be against selling one of their most promising talents, but could be tempted at the right price.