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KYIV, Ukraine—Drones struck two commercial ships sailing just outside Bulgaria’s territorial waters on Tuesday, posing a fresh test for NATO as the alliance confronts a growing number of suspected Russian attacks at its land and sea borders.

The North Atlantic Treaty Organization is pressing its members to call out Russia for such attacks when there is enough evidence to point to the Kremlin’s involvement, while seeking to avoid a direct confrontation between the military alliance and Moscow.

Tuesday’s strikes sank one of the vessels and left 11 people missing, according to officials in Bulgaria.

NATO Secretary-General Mark Rutte said that he had spoken with Bulgarian Prime Minister Rumen Radev but didn’t apportion blame for the strikes.

Radev summoned the Russian ambassador over the incident, the second such assault on commercial shipping in the area in a week. In both cases, Radev said that there was no conclusive evidence to blame a specific actor but that logic pointed to Russian involvement.

“The outcome benefits the Russian Federation,” he wrote on social media. Ukraine blamed Russia for the attack, which Radev said struck a Ukrainian-owned vessel, the Chelsea 3, with three aerial drones.

Russia’s Ministry of Defense didn’t immediately respond to a request for comment.

The spate of attacks abutting NATO’s maritime borders marks a new phase in the conflict in the Black Sea, one of the world’s most important shipping corridors, where this summer, Russia and Ukraine set each other’s vessels and port infrastructure ablaze in competing economic-pressure campaigns.

As a defensive alliance, NATO plans—for now—to let its member states take the lead on the response to attacks that don’t encroach directly on its territory, NATO officials said, rather than retaliating to them as a bloc.

The approach is designed in part to avoid the alliance being lured into triggering Article 5, which would compel its members to come to the aid of the one under attack and risk dangerous escalation. Triggering the mutual-defense pact but not responding forcefully would undermine alliance deterrence and make it look weak in Kremlin eyes, NATO officials fear.

Still, the latest attacks have presented a test of how restrained the alliance is prepared to be as European leaders, who have accused Russia of increasingly brazen attacks on their soil, now contend with strikes on ships skirting their waters.

Last month, Germany said Russia had sent an explosive drone to Leipzig airport , a transit hub for supplies to Ukraine. Russia denied involvement. The incident fell short of triggering a military response by NATO.

“Russia’s drone attacks on commercial ships off in the Black Sea are the visible edge of a widening shadow war across Europe,” said Martin Vladimirov , director of the geoeconomics program at the Center for the Study of Democracy, a European public-policy institute.

NATO airspace in Europe has become an active front line this year, as drones detected over its territory have been shot down using NATO jet fighters. More than a dozen members of the alliance reported airspace incursions or finding remnants of drones on their territory this summer.

“Russia will expand its hybrid warfare not necessarily in preparation for an invasion but in order to undermine NATO’s unity and the support for Ukraine,” Vladimirov said.

Rutte said the bloc would continue to work closely with allies in the Black Sea. “We have already strengthened our posture along the eastern flank and will continue to monitor the security environment, adjusting our deterrence and defense posture as needed,” he wrote on X.

The Tuesday attack saw three aerial drones strike the deck and side of the Malta-flagged Chelsea 3, some 18 nautical miles east of the Bulgarian shoreline, Radev said. A video released later by authorities showed how, just before dawn, a Bulgarian border-police vessel pulled alongside the stricken vessel, battling rough seas. All 15 crew members, seven Ukrainian and eight Azerbaijani nationals, were transported to safety in the Bulgarian port of Varna, Radev said.

Another vessel, HS Dignity, which was struck 17 nautical miles east of the Bulgarian shore, sank, with five people rescued and 11 crew members still missing, authorities said.

A week earlier, aerial and maritime drones hit two merchant ships in Bulgaria’s exclusive economic zone, according to Radev. A Togo-flagged vessel sank with no crew found by the rescuers, while a Palau-flagged vessel was evacuated after a fire broke out.

That came a day after another ship was attacked in Romania’s exclusive economic zone, with two people reported killed and 11 more rescued, the NATO member’s officials said. Ukrainian President Volodymyr Zelensky said the Turkish-owned vessel was transporting corn and came under attack from two Russian drones.

In his address to the Warsaw Security Forum on Monday, he warned European leaders about the persistent threat of hybrid attacks for their countries.

“There is no such thing as a small act of destabilization. Our intelligence services know that Russia is planning many aggressive hybrid operations against Europe,” he said, urging unity in defense.

Ukraine has maintained that it is ready for a ceasefire in the Black Sea, which would reopen its lifeline to global markets. Zelensky said Kyiv discussed various options to protect food security and safe navigation with Egypt, India and Turkey.

“Ukraine is ready to do its part to ensure calm, safe shipping and food security in the Black Sea if Russia is ready to take real steps toward de-escalation, too,” the Ukrainian president said in late September. “We have to start somewhere—and Russia has to start by respecting others.”

Write to Anastasiia Malenko at anastasiia.malenko@wsj.com and Georgi Kantchev at georgi.kantchev@wsj.com