The centre-half will spend the rest of this season with the Serie A club, who have an option to make the deal permanent next summer.
Roma say they will pay an initial £1m loan fee and a total price of £5.5m for any full-time transfer.
France international Yanga-Mbiwa was brought in from Montpellier during the 2013 January transfer window and went on to make 11 Premier League starts that season.
But Mike Williamson and Fabricio Coloccini are now established as Alan Pardew's first-choice pairing, and Yanga-Mbiwa is yet to play a minute's football this term.
On signing for his new club he said: "I am happy, forza Roma."