The 29-year-old visited United’s headquarters on Thursday to define his exit and is expected to fly to Italy in the evening to undergo a medical.
Talks between the two clubs started on Tuesday with United initially requesting €20m (£18m) between a loan fee and option to buy, but eventually decided to opt for a one-year loan to avoid long bureaucratic matters that could have scuppered the move.
The defender was omitted from United’s squads for their first three Premier League fixtures, including their 4-0 win against Chelsea and their 1-2 defeat to Crystal Palace.
Roma are keen on bringing in an experienced centre-back after seeing efforts to land Tottenham’s Toby Alderweireld, Liverpool’s Dejan Lovren and Juventus’ Daniele Rugani failing to materialise this summer.
The Serie A giants have seen stalwart Kostas Manolas leave for Napoli for €36m this summer but need more cover after bringing in Italy Under-21 defender Gianluca Mancini from Atalanta for €20m.