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Leeds have been linked with the 22-year-old for a number of weeks as Massimo Cellino looks to give head coach Dave Hockaday an improved squad for the upcoming Championship season.
While the majority of Leeds fans would want a permanent deal, the Yorkshire Evening Post claimed last week that any deal would be on a temporary basis.
The Roma midfielder has now been allowed to leave the club and the Italy U21 international was seen on Tuesday night outside Field Mill where Leeds played Mansfield in a pre-season friendly.
Viviani was captain of Roma's Primavera side last season and he scored five goals in 27 games while on loan at Pescara and Latina.
He is set to become the club's latest signing of the season as the major overhaul of the squad continues at Elland Road.
The Whites have already signed Gaetano Berardi, Souleymane Doukara, Marco Silvestri, Stuart Taylor and Tommaso Bianchi.
Viviani is well known for his free-kick taking ability but it remains to be seen if he will be able to handle the physicality of the English game.
United's new signings will have a less than three weeks to settle in with the first game against Millwall coming up on August 9.