In televised comments on Monday, Lavrov said that Moscow could supply more evidence of what it has said was a long-planned Ukrainian plot to destabilize Crimea.
Moscow accused Kiev last week of planning terrorist attacks in Crimea to provoke a new conflict over the region which seceded from Ukraine and reintegrated into Russia in 2014.
Lavrov repeated those claims after talks with German Foreign Minister Frank-Walter Steinmeier in the Urals city of Yekaterinburg but also extended an olive branch to Ukraine.
Lavrov said he did not think anyone was interested in cutting diplomatic ties between Russia and Ukraine yet.
Steinmeier said all sides should refrain from doing anything that would escalate the already tense situation.
On Thursday, Russian President Vladimir Putin accused Ukraine of “practicing terror.”
“This is very alarming news. In fact, our security services prevented an incursion into the territory by a sabotage-reconnaissance group from Ukraine's Defense Ministry,” Putin said.
He later held a meeting with his security chiefs to discuss “additional measures for ensuring security for citizens and essential infrastructure in Crimea.”
Russia's Federal Security Service said it had thwarted an incursion by the Ukrainian military into Crimea, saying two Russians were killed in the incident.
People in Ukraine’s Black Sea peninsula of Crimea voted for rejoining the Russian Federation in a referendum in March 2014. The move angered the West which branded it as Moscow’s annexation of the territory.
The United States and its allies in Europe accuse Moscow of having a hand in the conflict in eastern Ukraine where residents seek secession.
The crisis in eastern Ukraine has left nearly 9,500 people dead and over 21,000 others injured, according to the United Nations. Sporadic fighting between Kiev troops and pro-Russia forces continues to claim more lives despite a ceasefire.
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