Romanian President Klaus Iohannis condemned Russia's continued attacks on Ukrainian infrastructure "in the proximity of Romania" as unacceptable.
Ukraine's air force said during the night that Russian drones were heading for the Danube river, where Ukraine has two ports, Izmail and Reni.
Ukraine's military command in the south said air defences had been operating for almost three hours.
Odesa regional leader Oleh Kiper said emergencies services were working on the site of the latest Russian attack, and there were no reports of any casualties.
President Volodymyr Zelensky said that "unfortunately there has been damage" - and the regional chief posted several images on social media indicating that several structures had been hit.
Ukraine's defence ministry said an elevator had been hit - and officials said the Izmail district prosecutor had launched an investigation into a cargo terminal, a warehouse and an elevator that were all damaged, without detailing exactly where in the Odesa region.
Last week Russian drones also attacked grain warehouses in Reni, further up the River Danube and also next to Romanian territory.
President Iohannis said on Wednesday that the attacks so close to Romania were war crimes that further affected Ukraine's "capacity to transfer their food products towards those in need in the world".
Russia had earlier attacked the big Black Sea ports of Odesa and Chornomorsk, where authorities said 60,000 tonnes of grain were destroyed.
When it pulled out of the grain deal in July, Russia threatened to target any vessels heading towards Ukraine's Black Sea ports, essentially imposing a naval blockade.
Ships have continued to sail across the Black Sea to Ukraine's ports on the Danube, which has grown in importance as an alternative.
Ukraine is one of the world's major exporters of wheat and corn, and the bulk of the shipments had been moving from the country's Black Sea ports.
Now that Ukraine is unable to use its main Black Sea ports, the smaller-capacity Danube ports have become pivotal for its exports.
Wheat prices spiked on world markets immediately after the Russian withdrawal from the grain deal.
There are now also concerns about global food security, especially for impoverished African and Asian nations.
Overnight, Russia also launched more than 10 drones against Ukraine's capital Kyiv, local officials say.
All the projectiles were destroyed by anti-aircraft systems but several non-residential buildings were damaged by falling debris, the officials say.
Russia has so far not publicly commented on the reported attacks.