Four days after the worst fighting in more than a decade broke out between the Southeast Asian neighbours, the death toll stands above 30, mainly civilians.
More than 130,000 people have been evacuated from border areas in the two nations.
Cambodia's Defence Ministry said Thailand had shelled and launched ground assaults on Sunday morning local time at multiple points, including in Phnom Kmoach, which borders Thailand's coastal Trat province.
The ministry's spokesperson said heavy artillery was fired at temple complexes.
The Thai army said Cambodia had fired shots into several areas, including near civilian homes, early on Sunday.
The governor of Surin told Reuters artillery shells had been fired into the province, damaging a house and killing some livestock.
In the Thai province of Sisaket, Reuters reporters heard shelling early on Sunday and said it was unclear which side of the border it was on.
"If there is a ceasefire, things will be better," Sisaket resident Thavorn Toosawan told Reuters. "It's great that America is insisting on the ceasefire because it would bring peace."
Trump said on Saturday he had spoken with the prime ministers of Thailand and Cambodia and they had agreed to meet immediately to quickly work out a ceasefire to end fighting that began on Thursday.
Bangkok and Phnom Penh each say the other side started the hostilities.
"Both Parties are looking for an immediate Ceasefire and Peace," Trump wrote on social media.
Cambodian Prime Minister Hun Manet endorsed the call for the fighting to stop. "I made it clear to Honourable President Donald Trump that Cambodia agreed with the proposal for an immediate and unconditional ceasefire between the two armed forces," Hun Manet posted on Facebook, noting he had also agreed to Malaysia's earlier ceasefire proposal.
Thailand's response was more qualified, as it had been with the proposal from Malaysian Prime Minister Anwar Ibrahim, saying Cambodia needed to do more before talks could begin.
"I thanked President Trump for his concern and expressed that Thailand agrees in principle to have a ceasefire in place. However, Thailand would like to see sincere intention from the Cambodian side," acting Prime Minister Phumtham Wechayachai said on Facebook.
The nations have faced off since the killing of a Cambodian soldier late in May during a brief skirmish.
Troops on both sides of the border were reinforced amid a full-blown diplomatic crisis that brought Thailand's fragile coalition government to the brink of collapse.
Thailand and Cambodia have bickered for decades over undemarcated points along their 817-kilometre land border, with ownership of the ancient Hindu temples Ta Moan Thom and the 11th-century Preah Vihear central to the disputes.
Preah Vihear was awarded to Cambodia by the International Court of Justice in 1962, but tension escalated in 2008 after Cambodia attempted to list it as a UNESCO World Heritage site, and skirmishes over several years brought at least a dozen deaths.
Cambodia said in June it had asked the court to resolve its disputes with Thailand, which says it has never recognised the court's jurisdiction and prefers a bilateral approach.
