"50 to 60 people have been arrested in Kashan's illegal gatherings", State-run Mehr News Agency quoted the Kashan prosecutor Ahmad Takbirgo who added "Any kind of illegal gathering or arrogant activists in Kashan will be treated in lawful and serious manner".
Nine people were killed across Iran overnight amid a wave of anti-government protests, state television reported on Tuesday, while another media report said 450 protesters have been arrested.
State TV said armed demonstrators had tried and failed to seize police and military bases.
Rouhani was elected to a second four-year term last year with a pledge to strengthen the economy, while the country's supreme leader, Ayatollah Ali Khamenei, has presided over Iran since 1989.
After the 2009 Iranian presidential election, hundreds of Iranians took to the streets calling for the removal of former President Mahmoud Ahmadinejad from office.
The Asharq Al-Awsat newspaper reported that the law enforcement agencies had arrested almost 400 people in Iran, including 200 in Tehran, for taking part in the protests.
Turkey on Tuesday said it was concerned by reports of people dying and public buildings being damaged. "It's unclear how this is happening from town to town, except through social media". Rouhani finally broke his silence on Sunday night about the protests that mark the biggest test for the regime since the 2009 "Green Movement".
He blamed the US, Israel, and Saudi Arabia for involvement in the unrest in order to destabilize the country.
Iran is a major OPEC oil producer and regional power but frustrations have grown at home - where youth unemployment reached 28.8 percent previous year - while the Islamist regime has pursued its regional ambitions in Iraq and Syria.
A Monday report by the semi-official ILNA news agency quoted Hedayatollah Khademi, a representative for the town of Izeh, as saying the deaths happened there Sunday night.
Regional governor Mostafa Samali told Fars that only one person was killed in an incident unrelated to the protests, and the suspected shooter had been arrested.
US President Donald Trump, who has been tweeting in support of the protesters, continued into the New Year, describing Iran as 'failing at every level despite the bad deal made with them by the Obama Administration'. 'They are hungry for food & for freedom. Those foreign interventions are resented by many Iranians who want their leaders to create jobs instead of engaging in costly proxy wars.
German Foreign Minister Sigmar Gabriel also said in a statement that "after the confrontation of the past days it is all the more important for all sides to refrain from violent action".