As the sanctions bite, Iran's oil exports have dropped from 2.5 million barrels per day (bpd) in April last year to around 400,000 bpd in May.
Last week, an Iranian lawmaker said Zanganeh and Rouhani had fought over the issue during a cabinet meeting.
"I have had no problem with Mr Rouhani... I do not know who is giving the lawmakers such news," Zanganeh was quoted as saying by the oil ministry's SHANA news agency.
"I will not retreat," Zanganeh said, in an apparent reference to mounting pressure on him to quit.
The United States reimposed sanctions last year on exports of Iranian oil after President Donald Trump unilaterally pulled out of a 2015 accord between Iran and six world powers to curb Tehran's nuclear programme.
Last month, a separate Iranian lawmaker said he was gathering signatures in the parliament to support a motion to oust Zanganeh for his failure to counter the US sanctions on Iran's oil sales. He has failed so far to gather enough signatures to move the motion forward.
On Monday Iran said it would breach internationally agreed curbs on its stock of low-enriched uranium in 10 days, in a move likely to worsen already high tensions with Washington, though it also said European nations still had time to save the deal.