Ukraine has denied Russia's claim of attempted armed incursion into peninsula, annexed by Moscow in 2014.
Russia says it has thwarted two attempts by armed Ukrainians to get into Crimea.
The country’s FSB security service says there have been at least two armed clashes recently on the border between Crimea and Ukraine.
It also says it has dismantled a spy network inside the annexed peninsula.
What has the FSB said?
The FSB says it thinks Ukrainian special forces had been planning attacks targeting critical infrastructure.
An FSB employee and a Russian soldier were killed in the clashes at the weekend, according to officials.
The FSB says it tackled one group of what it describes as Ukrainian “saboteurs” in the early hours of Sunday, smashing what it says was a Ukrainian “spy network”.
Ukrainian and Russian nationals were arrested.
Twenty homemade explosive devices, ammunition, mines, grenades and weapons normally used by Ukrainian special forces were recovered, the FSB said, adding that there were more incidents late on Sunday and early on Monday.
“Ukrainian special forces units tried to break through two more times with groups of saboteur-terrorists but were thwarted by FSB units and other forces,” a statement said.
“The aim of this subversive activity and terrorist acts was to destabilise the socio-political situation in the region ahead of preparations and the holding of elections,” the FSB said in a statement.
“I think it is obvious for everyone now that the current Kyiv authorities are not seeking a way to solve the problems through talks but choose terror. This is very concerning,” said Russian President Vladimir Putin.
What has Ukraine said?
A spokesman for Ukrainian military intelligence has dismissed the claims as “false information”.
Kyiv has denied Russia’s claim of attempted armed incursion into Crimea.
The Ukrainian Defence Ministry said in a statement that the FSB’s assertions look like an attempt to justify “acts of aggression” and the redeployment of military units to Crimea.
“Representatives of the Russian special services are trying to divert the attention of the local population and the international community from criminal acts to transform the peninsula into an isolated military base,” it said.
“Russian accusations towards Ukraine of terrorism in the occupied Crimea sound as preposterous and cynical as the statements of the Russian leadership about the absence of the Russian troops in the Donbass (region of Ukraine),” said Ukrainian President Petro Poroshenko.
What is the context?
Observers say there has been a recent increase in Russian military activity in northern Crimea.
Fighting has also been heavier in eastern Ukraine, where Ukrainian government troops are battling pro-Russian separatists.
If true, these latest developments would be the most serious escalation on the contested peninsula since Moscow annexed it from Ukraine in 2014.
US and EU sanctions to punish Russia for annexing Crimea remain in place.
Moscow has made it clear it has no intention of handing the peninsula back to Ukraine.