The two Southeast Asian nations have each blamed the other for the clashes that started on Monday, and remain at odds on a diplomatic solution to the border tensions that have been simmering for months.
Thailand's foreign minister said in an interview on Tuesday that he saw no potential for negotiations, adding the situation was not conducive to third-party mediation, while a top adviser to Cambodia's Prime Minister Hun Manet told Reuters his country was "ready to talk at any time".
Speaking at a rally in Pennsylvania on Tuesday evening, Trump listed a number of wars that he claimed he helped stop, including between Pakistan and India, and Israel and Iran, before commenting on the renewed conflict in Southeast Asia.
"I hate to say this one, named Cambodia-Thailand and it started up today and tomorrow I am going to have to make a phone call. Who else could say I'm going to make a phone call and stop a war of two very powerful countries, Thailand and Cambodia."
Asked about the prospect of Trump intervening again, Cambodian government spokesperson Pen Bona said Phnom Penh's position remained the same, that it wanted only peace.
A Thai government spokesperson said he did not know if the prime minister had spoken to Trump, as he was in parliament.
Cambodia on Wednesday withdrew its athletes from the Southeast Asian Games in Thailand, citing safety reasons and serious concerns from the team's families.
Thailand's military said BM-21 rockets fired by Cambodian forces landed near the Phanom Dong Rak Hospital in Surin district on Wednesday morning, forcing the evacuation of patients and staff to a shelter.
It also said drones and BM-21 rockets and tanks were used at other border points, including near the contested Preah Vihear temple complex.
"Our forces destroyed an anti-drone position to the south of Chong Chom in order to support operations to clear Cambodian elements in a mango plantation that has been planted across the line of operations," the military said in a situation update.
Cambodia's military said Thailand had used artillery fire and armed drones to launch attacks in Pursat province, fired mortars into civilian residences in Battambang province, and Thai F-16 fighter jets had entered Cambodian airspace and dropped bombs near civilian areas.
Trump has previously spoken to leaders of both countries and been central to the fragile truce between them since five days of fighting in July, which killed at least 48 people and was the heaviest clashes between the two in recent history.
In July, Trump used the leverage of trade negotiations to broker a ceasefire.
Thai Foreign Minister Sihasak Phuangketkeow told Reuters on Tuesday he did not think the threat of tariffs should be used to pressure his country into talks.
Thailand in November suspended de-escalation measures that were agreed at an October summit in Trump's presence, after a Thai soldier was maimed by a landmine that Bangkok said was newly laid by Cambodia, which rejects the allegation.
Both countries have said they have evacuated hundreds of thousands of people from border areas.
As of Tuesday night, Cambodia's defence ministry said nine civilians had been killed since Monday and 20 seriously injured, while Thai officials said four soldiers had been killed and 68 had been injured.
