Martinelli is in protective custody after being detained Monday, accused of graft and spying on political foes at home.
Judge Edwin Torres ultimately said the case's complicated nature meant he needs more time to make a decision. Martinelli meanwhile remains in custody, according to Alejandro Perez, a member of the ex-president's legal team.
Panama's Supreme Court ordered his arrest in 2015 over accusations that he used public funds to illegally spy on telephone calls and emails of more than 150 prominent opponents.
Panama made a request for his extradition last year.
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The international police agency Interpol issued a "red notice" on the government's request last month.
The United States Marshals Service arrested Martinelli on the basis of that warrant and the extradition treaty that exists between the two countries.
Panamanian authorities have opened some 200 investigations into Martinelli's administration, according to Transparency International.
Allegations include accusations that he helped embezzle USD 45 million from a government school-lunch program, as well as other cases of extortion, bribe-taking, misappropriation of public funds and abuse of power.