Rui Pinto is wanted in his native Portugal for attempted extortion, illegal access to data and other alleged crimes connected to the release of secret information about the financial dealings of clubs.
His lawyers appealed against the ruling by the Budapest Metropolitan Court.
Pinto said he feared for his own life and his family's.
"(It's) a matter of life or death," he said. "I ask you, please don't send me back to Portugal."
Football Leaks began as a website in 2015 revealing details of player salaries and transfer payments, typically involving Portuguese and Spanish teams and agents.
German magazine Der Spiegel and other European media started publishing documents from the latest batch in November with allegations including that European clubs planned for a breakaway league and Manchester City had misled UEFA to comply with financial monitoring rules.
Clubs have not denied the authenticity of the documents, while Pinto and media using the information have denied they were obtained by hacking.
He told Judge Judit Csiszar that he and his family in Portugal have received death threats and Portuguese authorities offered him no protection even though there are European laws to protect whistleblowers.
Pinto said he has been a "marked man" since Portuguese media revealed his identity.
"Unfortunately, I can't trust the Portuguese authorities," said Pinto, who spoke with reporters during a break in court proceedings and after the ruling was issued.
He said they had shown "a clear apathy" when it comes to investigating clubs and have been "completely biased" in cases involving football.
"A lot of European investigations were opened thanks to the revelations brought by Football Leaks and the medias involved in the Football Leaks," Pinto said in the court hallway, closely guarded by Hungarian police.
"At the moment, I think at least nine or 10 European countries are with me. The exception is Portugal, and that explains everything."
Pinto told the court he was ready to collaborate with any authorities and had been in touch with French prosecutors who, he said, had offered him witness protection if he testified.
He said he had also been contacted by US authorities about his findings, but could not give details.
If Pinto is extradited, Portuguese authorities will receive a large cache of computer and communications equipment - including external hard drives, memory cards, a laptop and mobile phones - confiscated by Hungarian police when they arrested Pinto at his Budapest home.
He has been living in the Hungarian capital since early 2015.
David Deak, Pinto's lawyer, presented several arguments in favour of rejecting the extradition request, including the fact there was no Portuguese arrest warrant when Pinto was detained in January by Hungarian authorities, only a European one.
An appeals court is expected to issue a ruling within a few weeks. If the extradition is upheld, Portuguese authorities will have 10 days to take over custody of Pinto.