The 24-year-old was on loan at Valencia last season and is surplus to requirements at Atletico Madrid.
Vietto was signed for £18million after impressing at Villarreal in 2014/15 but has struggled to repeat the form that saw him score 18 goals in 48 games there.
Vietto featured four times and scored twice for Atletico during their recent pre-season International Champions Cup tour and was awarded man-of-the-match after the 1-1 draw against Arsenal.
Fulham are set to have an incredibly busy deadline-day, with Montoya, Joe Bryan and Zambo Anguissa all set to join the club today.
Fulham's Premier League opener is at home to Crystal Palace on Saturday.