The 34-year-old former Juventus forward, who was born in France to Argentine parents, underwent medical tests in Buenos Aires on Friday. He expects to finalize the deal on Monday.
River was relegated to the second division for the first time in its 110-year history last season.
Although Trezeguet is well past his prime, the signing still represents a publicity coup for River, which has won more domestic championships than any other club in Argentina. River has won 33 first-division titles, 10 more than archrival Boca Juniors.
River must finish in the top two at the end of the 38-game season in the Primera B Nacional to automatically return to the top flight. The club is currently second after 18 matches, two points behind leader Instituto de Cordoba.
''I think that in this difficult time for River, to be able to give an important hand at this moment could be good and very interesting for the club,'' Trezeguet said in an interview published on River's official website, accompanied by pictures of the striker touring the Bombonera stadium with club president Daniel Passarella.
''I want to form part of the history of this club and I think everything is in place to do it,'' Trezeguet added. ''There's just one objective and that's getting promoted to the first division.''
The striker has also played for Monaco in France and scored the winning 'golden goal' in the European Championship final against Italy in 2000, two years after helping Les Bleus win the World Cup as the host nation.
He spent his childhood in Argentina and came through the ranks at Platense.
After leaving Spanish club Hercules in the summer, Trezeguet has been playing in the United Arab Emirates with Baniyas.