There was no immediate reaction from Kyiv to the allegation.
The alleged target was Tikhon Shevkunov, 66, whom Russian media have described for years as "Putin's confessor".
He has maintained a public acquaintance with Putin since the late 1990s and the Kremlin has said the two men know each other well.
The Kremlin condemned the alleged assassination plot.
"It is clear that the Kyiv regime does not shrink from anything," spokesman Dmitry Peskov told reporters on Friday.
"In this case, nothing is sacred. This is once again confirmed."
In 2023, Shevkunov was appointed to the senior church rank of metropolitan of Crimea, the peninsula which Russia annexed from Ukraine in 2014.
In a statement, the FSB said it had detained two men it said had been recruited by Ukrainian military intelligence on the Telegram messenger service in mid-2024.
The FSB said it seized an improvised explosive device and two fake Ukrainian passports.
One of the men, Nikita Ivankovich, is a Russian cleric at a church in Moscow.
The other, Denis Popovich, a Ukrainian born in the western city of Chernivtsi, worked as Shevkunov's assistant.
The authorities released videos of both men confessing to the plot.
They spoke hesitantly and it was not clear under what circumstances the confessions had been obtained.
Popovich said he had been recruited to monitor Shevkunov's movements and threatened with the murder of his relatives unless he complied.
He said he was then tasked with finding an accomplice in order to "eliminate" Shevkunov.
He said the plan was to plant a bomb in a residential building of the 14th-century Sretensky Monastery in Moscow, where Shevkunov was superior until 2018.
Ivankovich, in his confession, said the men had been supplied with a home-made bomb to carry out the attack.
The video showed investigators recovering a box and fake passports from a hiding place in a wood.
Ukraine has taken responsibility for a number of assassinations in Russia since the start of the war - most recently for the killing of Igor Kirillov, the head of Russia's Nuclear, Biological and Chemical Protection Troops, in December.
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