The report states that Roma are keen to bring in the 25-year-old as a replacement for the ageing Maicon.
Last week Turkish-football.com confirmed that Galatasaray were in talks with the right-back who is expected to leave United following their acquisition of Matteo Darmian from Torino
The Brazil international current contract has just one year left to run and Roma believe he could be available for a cut-price.
TalkSport go onto claim that United are holding out for a fee in the region of £5 million.
Rafael is expected to miss United’s pre-season USA tour which begins today.
Galatasaray have made strengthening at right-back a priority over the summer transfer window after Tarık Çamdal failed to settle in last season.
Rafael would become the Lions third Brazilian player should he join; Felipe Melo and Alex Telles being the other two.