Trump said on Sunday the US may meet Iranian officials and he was in contact with Iran's opposition while piling pressure on its leaders, including threatening possible military action over lethal violence against protesters.
US-based rights group HRANA said it had verified the deaths of 572 people - 503 protesters and 69 security personnel, with 10,694 people arrested since the protests began on December 28 and spread around the country.
Reuters was unable to independently verify the tallies.
The flow of information from the Islamic Republic has been hampered by an internet blackout since Thursday.
Iran's leaders, their regional clout much reduced, are facing fierce demonstrations that evolved from complaints about dire economic hardships to defiant calls for the fall of the deeply entrenched clerical establishment.
But despite the massive scale of the protests, there are no signs of splits in the Shi'ite clerical leadership, military or security forces, and demonstrators have no clear central leadership.
The opposition is fragmented.
In verified video footage, Iranians gathered at the Kahrizak Forensic Centre in Tehran on Sunday, standing over rows of dark body bags.
Iranian authorities have not given an official death toll but blame the bloodshed on US interference and what it calls Israeli- and US-backed militants.
State-run media has focused attention on the deaths of security forces.
Iran's Ministry of Intelligence said on Monday it had detained "terrorist" teams responsible for acts including killing paramilitary volunteers loyal to the clerical establishment, torching mosques and attacking military sites.
Iranian Foreign Ministry spokesman Esmaeil Baghaei said "contradictory messages" from the US showed a lack of seriousness but that contacts continued.
"The communication channel between our Foreign Minister Abbas Araqchi and the US special envoy (Steve Witkoff) is open and messages are exchanged whenever necessary," he said, adding that contacts also remain open through traditional intermediary Switzerland.
Araqchi reiterated in a briefing to foreign ambassadors in Tehran that the Islamic Republic was ready for war but also open to dialogue.
He said "the situation has come under total control" and that the internet services across the country would be restored shortly.
"The internet will return soon in co-ordination with security authorities," he said, according to statements reported by al-Mayadeen.
Addressing a large crowd in Tehran's Enqelab Square on Monday, parliament Speaker Mohammad Baqer Qalibaf said Iranians were fighting a war on four fronts - "economic war, psychological warfare, military war against the US and Israel, and today a war against terrorism".
Araqchi said on Monday that a total of 53 mosques and 180 ambulances had been set on fire since the protests erupted.
CCTV footage from inside Tehran's Abuzar Mosque showed a dozen people, most wearing face masks, ransacking the structure, throwing books onto the ground and destroying furniture last week.
Reuters verified the time stamp and location.
State media reported that the mosque was set on fire on January 9.
Trump said on Sunday that Iran had called to negotiate on its disputed nuclear programme.
Israel and the US bombed Iranian nuclear sites in a 12-day war in June.
"Iran wants to negotiate, yes. We might meet with them. A meeting is being set up but we may have to act because of what is happening before the meeting, but a meeting is being set up. Iran called, they want to negotiate," he told reporters on Air Force One.
with AP and EFE
