Saviola, 25, has already confirmed he will leave the Spanish champions when his contract expires this summer, meaning he can move for no transfer fee.
According to The Sun, the Argentine has made it clear Old Trafford is his preferred destination after three years at Barca, despite interest from Arsenal, Liverpool, Bolton, Valencia and Real Madrid.
He said: "I consider my career at Barcelona is finished. There are several offers on the table."
Saviola moved to Spain from River Plate in 2001 but despite an impressive goalscoring ratio, he never really established himself in the first-team.
He was shipped out on loan to Monaco for the 2004-05 season and then to Sevilla last year where he helped the club win the UEFA Cup.
Earlier this season he scored 8 goals in 6 matches in the absence of star striker Samuel Eto'o, but again found himself benched when the Cameroonian returned.
United have been linked with a host of strikers since the end of the season including Eto'o, Fernando Torres and Nicolas Anelka.