The offer to suspend attacks was transmitted by Ukraine via a third party and it was still waiting for a response, the source who is familiar with the matter told Reuters.
Both Russia and Ukraine, major players in the world agriculture market, have accused each other of intensified attacks on vessels used for exports.
Ukraine was forced to turn to alternative shipment routes when many shipowners halted stops at ports in late July in the southern region of Odesa - a key hub for grain exports - wary after Russian strikes on dozens of ships.
For its part, Russia had to suspend operations at all three terminals at its Black Sea port of Novorossiysk on Wednesday and Thursday after a Ukrainian attack, and will have to cut its grain exports further.
The Kremlin and Russian foreign ministry did not immediately respond to a request for comment on the report of Ukraine's offer.
Before the report, Deputy Russian Foreign Minister Alexander Grushko said Russia had received no formal Black Sea ceasefire proposal.
"Recently, we have been hearing many calls for various kinds of moratoriums and truces. These ideas are being put forward through various channels but we have not received any formal proposals," he said, according to Russia's state news agency TASS.
Euronext wheat pared gains in choppy trading on Thursday to come off a two-week high following the Reuters report.
Russia has repeatedly aimed to block Ukrainian port operations and shipments that are key to Ukraine's war-ravaged economy.
Following Russia's 2022 invasion, the United Nations and Turkey brokered a deal allowing Ukrainian grain exports to continue to stave off a looming food crisis.
In 2023, Russia refused to prolong the agreement.
After that, Ukraine established another sea route that had remained operational until the latest round of escalation.
Alternatives - both rail and via the Danube - are extremely limited, Ukraine says.
On Sunday, Turkey conveyed its concerns about attacks to Russia and Ukraine, saying both should declare a moratorium on attacks in the Black Sea.
The current de facto blockade of Black Sea ports has sent Ukrainian grain exports tumbling 76 per cent year-on-year so far in August, with the agricultural sector warning of vast consequences for the economy should it persist.
