The Israeli foreign ministry did not respond to a request for comment on the allegation by the Freedom Flotilla Coalition, an international non-governmental group.
The Maltese government said the vessel and its crew were secured in the early hours of Friday morning after a nearby tug assisted with firefighting operations.
Turkey's foreign ministry said Turkish nationals were on board at the time of the incident and it was working with Maltese authorities to transfer them to a safe location.
"We condemn in the strongest terms this attack on a civilian ship," it said, noting there were "allegations that the ship was targeted by Israeli drones".
"All necessary efforts will be made to reveal the details of the attack as soon as possible and to bring the perpetrators to justice," it said.
A ceasefire between Israel and Hamas broke down in March, with both sides blaming each other, and Israel sent troops back into Gaza and resumed airstrikes.
The NGO published video footage, filmed in darkness, showing a fire on one of its ships, the Conscience. The footage showed lights in the sky in front of the ship and the sound of explosions could be heard.
"Israeli ambassadors must be summoned and answer to violations of international law, including the ongoing blockade (of Gaza) and the bombing of our civilian vessel in international waters," it said.
The Maltese government said maritime authorities had received a mayday call shortly after midnight local time from a vessel outside of territorial waters, with 12 crew members and four civilians on board, reporting a fire.
It said a nearby tug headed to the scene and launched firefighting operations and a Maltese patrol vessel was dispatched. After several hours, the vessel and its crew were secure, it said, adding that crew had refused to board the tug.
A spokeswoman for the NGO, Caoimhe Butterly, said the attack took place as the ship was preparing for activists to board from another vessel. A transfer at sea had been planned rather than the ship going to harbour, for bureaucratic reasons, she said.
The coalition said it had been organising a non-violent action under a media blackout in order to avoid any potential sabotage.
The Gaza war started after Hamas-led fighters killed 1200 people and took 251 hostages to Gaza in the October 7, 2023 attacks, according to Israeli tallies. Since then, Israel's offensive on the enclave killed more than 52,000, according to Palestinian health officials.
Since March 2, Israel has completely cut off all supplies to the 2.3 million residents of the enclave, and food stockpiled during a ceasefire at the start of the year has all but run out, according to international aid agencies.
Israel accuses the Hamas militants who have run Gaza of exploiting aid - which Hamas denies - and says it must keep all supplies out to prevent the fighters from getting it.
Another coalition ship on a similar mission to Gaza in 2010 was stopped and boarded by Israeli troops, and nine activists died. Other ships have similarly been stopped and boarded, without loss of life.
Hamas issued a statement about the incident off Malta, accusing Israel of "piracy" and "state terrorism".
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