A spokesperson for the regional administration confirmed there had been blasts in Odessa but gave no further details.
He did not say whether there had been any casualties.
Oleksiy Honcharenko, a Ukrianian lawmaker from Odessa, said the city appeared to have come under attack in what he described as "revenge for our morning shelling of oil rigs near Crimea".
Sergei Aksyonov, who was installed by Russia as head of the Crimea region that Russia annexed in 2014, said drilling platforms owned by the Chernomorneftegaz energy company had come under attack in the Black Sea off Ukraine's south coast.
Three people were wounded, and a search was under way for seven workers, he said in a post on the Telegram messaging app.
"At this stage, 21 people have been rescued," Aksyonov told Russian state television, adding 109 people had recently been on the platforms.
He said there had been three attacks, with one leading to some injuries.
Russia's RIA Novosti news agency said the drilling platforms were located 71km from Odessa.
The Ukrainian military did not comment on the incident at first.
Reuters was unable to immediately verify the reports of the attack.
Ukraine's military said earlier on Monday that it had shot down at least one Russian missile near Odessa.
City authorities said 80 windows had been shattered by the blast when the missile was intercepted.
Russia did not immediately comment on the reports.
The platforms are located about 100km from the coast of Odessa province and 150 kilometres from the Crimean Peninsula on the Black Sea shelf.
In the course of Russia's annexation of Crimea, the installations were occupied by Russia's military in March 2014.
with reporting from DPA
