But the chances of a return to River's arch-rivals Boca Juniors for Tevez, who has been enjoying the Argentine summer playing golf while Manchester City study offers for their disaffected striker, are extremely remote.
Trezeguet arrived in Buenos Aires, the city where he grew up and forged his future as a footballer, on Thursday and was reported to be meeting with River President Daniel Passarella later in the day or on Friday.
"I'm in perfect shape to give River my best," Trezeguet told reporters on arrival at Ezeiza international airport.
"To play for this club and in Argentine football is to crown a dream. I left [Buenos Aires] very young and I'm really enthusiastic about this [move]," added the 34-year-old who helped France win the 1998 World Cup and 2000 European Championships.
River coach Matias Almeyda, a former Argentina midfielder who played at the 1998 and 2002 World Cups, is equally enthusiastic about having Trezeguet in his squad as he looks to get River promoted back to the first division.
"I've talked with Matias and we had a very good dialogue. He explained the team's situation to me. The club's objective is very clear, and his too," said Trezeguet, a childhood River fan who was born to an Argentine footballer father in Rouen, France in 1977.
"I love Trezeguet, he has the stamp of a goal-scorer," Almeyda, who was still a River player when the record 33 times Argentine champions were relegated in June, told the sports daily Ole on Wednesday.
Trezeguet, who played for Platense as a boy, began his European career at Monaco before moving to Juventus where he has already had experience of second division football after the Italian giants were relegated in the Calciopoli match-fixing scandal.
He has since played for Hercules in Spain and Baniyas in the UAE and his status as a free agent should ease his potential move to River.
This is far from the case for Tevez, tied to City by a contract until 2014 with a possible to move to one of the two big Milan clubs extremely difficult due to the Premier League's asking price and dislike for a loan deal.
Tevez has said he would like to play for Boca, his first big club, in the Argentine league and Copa Libertadores in 2012 telling the non-sports magazine Gente (People) recently he had asked his Argentine agent Adrian Rouco to try to sort out his release from City.
"Travel [to Europe], listen to all the offers and try to get it all disentangled. In January I want to be playing for Boca," Gente quoted Tevez as telling Rouco while he played golf in Cordoba province 800 km (500 miles) from the capital.
"It doesn't matter if I have to take a pay cut, but I'm tired of the travelling and the comings and goings... I want to do the pre-season [training] and play the Copa Libertadores."
Tevez won the Libertadores, South America's champions league, and the old Intercontinental Cup with Boca in 2003 before moving to Corinthians in Brazil and on to England where he played for West Ham United and Manchester United before joining City in 2009.
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