The online editions of leading newspapers O Globo and Folha de S.Paulo reported, without citing sources, that Battisti would not be extradited.
Earlier in the week Lula said he would decide by Friday. His aides said on Wednesday there was no formal decision yet.
The Supreme Court had ruled last year that Battisti should be extradited on murder convictions from the 1970s in Italy, but it left the final decision to Lula.
Battisti denies the murders and says he is being persecuted politically in Italy.
He faces life in prison there for the murders in the 1970s, a violent period known as the “Years of Lead,” when he belonged to a guerrilla group called “Armed Proletarians for Communism.”
Lula, who ends his second term in office on Jan. 1, had granted him refugee status in January 2009, straining ties with Italy. The pending decision has been one of the main issues during the final days of Lula’s presidency.
Several senior members of his leftist Workers’ Party had supported Battisti’s cause and visited him in prison on the outskirts of the capital Brasilia.
Lula wanted to decide on the case in order to spare his successor, Dilma Rousseff, from having to wade into controversy early in office.
But some analysts say he waited until the very end of his term in order to avoid renewed diplomatic tensions with Italy.
Battisti escaped from an Italian prison in 1981 and lived in France for years, but fled when Paris approved his extradition in 2006. He was arrested on the run in Brazil.
