Xabi Alonso claims no Premier League offers were put to him before leaving Real Madrid for Bayern Munich.
Blackpool have signed Rochdale full-back Tom Kennedy on loan, while Tangerines striker Francois Zoko has joined Bradford.
Scocco only joined Sunderland in January but he played a limited role during the second half of last season and there has been a lot of speculation that he could return to Argentina.
River Plate are among the clubs to have been linked with a move for the 29-year-old and Gallardo says he is their No.1 target.
He told Superclasico950: "Our first priority is (to sign) Scocco."
River president Rodolfo D'Onofrio also confirmed the club’s interest in Scocco, saying that he has been on their radar for some time.
He said: "Gallardo asked us to sign a striker.
"We have been interested in Scocco since January, (then manager) Ramon Diaz wanted us to sign him and Francescoli recommended us to sign him.
"(Sporting director) Enzo Francescoli told us that Scocco is an ideal player for River."
According to reports in the Argentine press, River have offered around £4million to try and sign Scocco.