The Chilean has been left out of the Giallorossi's pre-season training camp in Austria amid speculation regarding his future.
The 32-year-old is expected to leave the Italian capital and River's vice-president Omar Solassi has said that the club are ready to make sacrifices in order to sign him.
"We are following David - he is a great player and one we want at River Plate," Solassi told Chilean newspaper La Tercera. "We will talk together and make an effort to bring him here.
"We have some problems regarding our spaces for foreign players, but to facilitate his arrival we are ready to make the necessary sacrifices."
Pizarro spent the latter part of last season on loan at Manchester City.