By Agence France-Presse
A Philippine patrol ship pursued six Vietnamese fishing boats 30 nautical miles off the northern coastal town of Bolinao, a regional military spokesman told AFP.
Officials are investigating the cause of the pair's death, but their bodies were found after a Vietnamese boat slammed into the Philippine vessel, said Lieutenant Jose Covarrubias.
The five fishermen in custody would face poaching charges as they were within Philippine waters, he added.
"The navy ship spotted the six fishing boats (which were) using bright lights to attract fish. While it was chasing the boats, one boat manoeuvred and hit our vessel," said Covarrubias.
"The officers boarded and searched the boat, resulting in the apprehension of the five (fishermen) and they found the two dead bodies."
A separate statement from the Philippine foreign affairs department on Monday said the Navy personnel had fired warning shots after the vessel had initiated "very dangerous manoeuvres", but Covarrubias said he could not confirm this.
Foreign fishermen have often been caught trespassing in Philippine seas. Last year President Rodrigo Duterte ordered the release of 17 Vietnamese fishermen caught in his nation's waters.
In 2013 the Philippines apologised to Taiwan over the killing of a Taiwanese fisherman by Filipino coastguards who said his vessel had illegally sailed into local waters.
That incident triggered a diplomatic spat, with Taiwan suspending the hiring of Philippine workers and recalling its envoy to Manila in protest.
On Monday, Foreign Secretary Alan Peter Cayetano said the Philippines would conduct a "fair and thorough" investigation into the deaths of the Vietnamese fishermen.
"We would like to offer our sympathies over the unfortunate loss of life," Cayetano told Vietnamese Foreign Minister Pham Binh Minh at a meeting on the sidelines of the United Nations General Assembly in New York.