The moderate Rouhani, who is expected to stand for a second four-year term in the May vote, has targeted the judiciary in a series of public statements over the case of a billionaire businessman on death row for corruption .
Judiciary chief Ayatollah Sadegh Larijani has hit back with accusations of his own, and on Sunday supreme leader Ayatollah Ali Khamenei made reference to the dispute.
In a speech to thousands of people in the religious city of Qom, Khamenei alluded to "recent arguments" among powerful figures, adding: "This will be resolved with the help of God."
Rouhani launched the first salvo in a speech late last month referring to the case of Babak Zanjani, sentenced to death for fraud and other financial crimes in March after a lengthy trial that saw him convicted of fraudulently pocketing $2.8 billion.
Zanjani was arrested in 2013, three months after Rouhani took office pledging a crackdown on corruption .
He rose to prominence after sanctions were imposed on Iran over its nuclear programme, boasting in media interviews of finding ways around banking restrictions to ensure Tehran continued to receive money for international oil sales.
In his speech, Rouhani appeared to suggest that the judiciary had not done enough against others potentially involved in the case or to recover funds embezzled by Zanjani.
"How is it possible that a single man can pocket $3 billion? Who has helped him? Who are his accomplices?" Rouhani asked.
The death sentence against Zanjani was "very well", Rouhani said, "but what becomes of the money?"
Last week Larijani hit back, defending the judiciary's handling of the case and criticising the government for not doing enough to help track down missing funds.
"It should not be implied that the judiciary has done little on this case," Larijani said, insisting it was working "to recover the misappropriated money".
"The case remains open... we have summoned former ministers and the former director of the central bank," he said.
Larijani, brother of the moderately conservative parliament speaker Ali Larijani, said the government and foreign ministry were "not doing their duty of tracing the money taken abroad by Zanjani".
Iranian authorities have so far only been able to recover about $600 million of the embezzled money from the seizure of Zanjani's assets in Iran.
Larijani called on the foreign ministry to demand information on the funds from the governments of Malaysia, Turkey, Tajikistan and China. The ministry replied that it had given its "full cooperation" to the judiciary.
Larijani also revealed claims made by Zanjani during his questioning that he had paid hundreds of thousands of dollars to presidential candidates in 2013.
The judiciary chief provided no names but said it would begin to investigate the alleged payments.
That claim appeared to raise the stakes in the dispute and prompted Rouhani to urge the judiciary disclose its own financial dealings.
"The government is ready to shed light on its accounts and expects the judiciary to do the same," Rouhani said Tuesday on Twitter, in an apparent reference to recent criticism of judiciary bank accounts managed directly by its chief.
"The judiciary accounts are transparent and anyone can view them," judiciary spokesman Gholamhossein Mohseni Ejeie said in response to the tweet, demanding Rouhani "take action" for transparency.
"They need to tell the public how much Mr. Rouhani spent on the (2013 presidential) campaign and where they money came from," Mohseni Ejeie said.
Some critics have accused Rouhani of attacking the judiciary to appeal to his moderate base and divert attention from a lack of immediate economic results from the nuclear deal.
The 2015 agreement between Tehran and six world powers, which saw sanctions lifted in exchange for curbs on Iran's nuclear programme, is expected to be a centrepiece of Rouhani's re-election campaign.
In his speech on Sunday, Khamenei appeared to urge the two sides to overcome their differences.
"The existence of an independent and courageous judiciary must be appreciated by everyone," he said. "What defeats the enemy in its objectives is to have a strong judiciary and a government that is both brave and can plan accurately."