Egyptian militants from the Sinai Province group have previously focused on attacks on Egyptian soldiers and police in the Sinai peninsula, killing hundreds since the army ousted President Mohammed Morsi in 2013.
Photographs distributed by the group, and which have not been independently verified, appeared to show a rocket heading towards the ship, and then setting it ablaze on impact.
Egyptian military sources said in a statement that a coastguard launch had exchanged shots with “terrorist elements”, causing it to catch fire, but there had been no loss of life.
The incident took place in waters off the town of Rafah, in north Sinai, which borders on the Gaza Strip.
Witnesses in Gaza said they had heard explosions and gunfire. The attackers were said by military sources to have fled after firing on the vessel.
The group, which pledged allegiance to the self-styled Islamic State in 2014, is the most lethal militant group in Egypt. Its recent high-profile attacks have triggered the drafting of a sweeping counter-terrorism law.
On 1 July, 100 militants and at least 17 members of the security forces were killed in a single day of clashes and attacks claimed by the group.