Authorities in Mlang town on the southern island of Mindanao said they were trying to convince the fishermen to hand over the 2.5-metre (eight-foot) long crocodile to them for safekeeping.
"Somebody told them that the local government would give rewards but it's not true. We exerted all means to convince them but we failed," said Hernand Dapudong, the town's disaster monitoring chief.
The fishermen said they had accidentally netted the crocodile in Mindanao's sprawling Liguasan marsh on Saturday. They then brought the creature, tied up with rope, into nearby Mlang on a motorboat.
Dapudong said the fishermen were asking 40,000 pesos (29,000 baht) for the crocodile, estimated to weigh 100 kilograms (220 pounds) but did not say what they would do to it if they were not paid.
It was not clear where the fishermen were keeping the animal while the wrangle continues.
Dapudong warned that the crocodile could die from stress or escape and harm residents.
Mlang's mayor Joselito Piñol has sought legal assistance from the Department of Environment and Natural Resources on what to do with the crocodile.
He is also asking other town officials to help, saying, "this is not about rewards but the safety of your constituents".
It is not yet known whether the reptile is a critically-endangered Philippine crocodile -- found only in the Philippines -- or the larger, more common saltwater crocodile which is not threatened and is found in several Asia-Pacific nations.