Ukraine almost never publicly claims responsibility for such attacks but has said destroying Russia's military infrastructure is crucial for its counteroffensive, Reuters reports.
Russian Foreign Ministry spokeswoman Maria Zakharova condemned the latest attacks on the bridge, saying they posed a danger to innocent lives and civilian infrastructure.
"There can be no justification for such barbaric actions and they will not go unanswered," she wrote on Telegram.
Separately, Russian forces destroyed 20 Ukrainian drones launched onto the Crimean Peninsula earlier on Saturday, Russia's Defence Ministry said.
There were no casualties and no damage as a result of the drone attacks, the ministry said on Telegram. It said 14 drones were destroyed by air defence systems and six were suppressed by electronic warfare.
Reuters could not independently verify the report. It was not clear what had been targeted in those reported attacks.
Drone attacks on Russian-controlled territories in Ukraine and deep inside Russia have increased since a drone was destroyed over the Kremlin in early May.
The Crimean Bridge was badly damaged last October in a powerful blast that Russian officials said was caused by a truck that blew up while crossing the bridge, killing three people.
Ukraine's SBU domestic intelligence agency later claimed responsibility for the sabotage operation.
