Fighting flared between Thai and Cambodian troops along the two countries’ disputed border early Sunday despite President Trump saying that the leaders of both countries had agreed to a cease-fire the previous day.

The continuing skirmishes show the challenge of resolving an increasingly bitter and long-running conflict, which has reached its deadliest point in at least 14 years. Analysts said that both countries could also be seeking to grab territory before an agreement is signed.

Thailand’s acting prime minister, Phumtham Wechayachai, said he told Mr. Trump that Thailand had agreed in principle to a cease-fire but would “like to see sincere intention from the Cambodian side.”

In a statement, Mr. Phumtham said he had asked Mr. Trump to convey to Cambodia that Thailand wanted to convene bilateral talks as soon as possible.

Cambodia’s prime minister, Hun Manet, said he hoped that Thailand would not go back on its cease-fire vow. He repeated his assertion that the Thais had violated a similar promise that the Malaysian prime minister, Anwar Ibrahim, had negotiated with both sides on Thursday.

Mr. Hun Manet said he had instructed his foreign minister to talk to Marco Rubio, the U.S. secretary of state, who would be working with Thailand’s foreign minister to help implement the cease-fire.

Thailand and Cambodia exchanged fire on their disputed border near the site of a temple, which is known as Prasat Ta Khwai to the Thais and Prasat Ta Krabey to the Cambodians. The deputy spokesman of the Thai army, Col. Richa Suksuwanont, said the first shots had been fired by Cambodia into Thai territory in several areas, including into civilian homes, early this morning.

Maly Socheata, a spokeswoman for Cambodia’s defense ministry, accused Thailand of still “invading aggressively,” saying that Thailand’s troops fired at around 2 a.m. on Sunday, shooting near Prasat Ta Krabey and also near another temple, an ancient one claimed by both nations, called Prasat Ta Moan Thom by the Cambodians and Prasat Ta Muen Thom by the Thais. That is the place where violence first erupted on Thursday.

At least 34 people have died so far in the conflict, which began after two months of tension over contested territory. The current toll has surpassed the last previous deadly conflict, from 2008 to 2011, which killed at least 15 people.