The Argentine side have yet to deposit an initial fee of $1.7million (£1.08m) for Gutierrez with Cruz Azul despite the Mexicanclub extending the payment deadline to 2 pm (1900 GMT) on Friday August 16.
"I've had no more contact with (River) and I know they haven't deposited anything," Cruz Azul's sporting director Agustin Manzo told reporters.
"There's no more deadline and the case will go to FIFA ...The player must come back (to Cruz Azul)."
The Mexican club later issued a statement confirming its approach to FIFA and asking for sanctions against River and Gutierrez.
Gutierrez joined Cruz Azul from Atletico Junior of Colombia last December but made it clear he wanted to a move to River in July as soon as the Argentine side's coach Ramon Diaz expressed an interest in signing him.
The 28-year-old former Racing Club striker has yet to make his debut for River, missing their first two matches of the "Inicial" championship because his international transfer approval has yet to come through.
However, Diaz told a news conference on Friday he had included Gutierrez in his squad for Sunday's visit to Godoy Cruzin Mendoza after being assured by the club management he would be able to play this weekend.
"We bought Teo because he is a player with a lot of class, atop goalscorer and we've been preparing him (for a debut) since he arrived (in Buenos Aires)," Diaz said.