MOSCOW, July 4 (Reuters) - A Russian lawmaker said on Thursday that the Russian deployment of missiles on Cuba was one option being looked at in response to what he cast as a U.S. escalation over Ukraine and an accelerating arms race among major nuclear powers.

President Vladimir Putin said last week that Russia should resume production of intermediate and shorter range nuclear-capable missiles and then consider where to deploy them after the United States brought similar missiles to Europe and Asia.

Sergei Mironov, the leader of the "Just Russia - Patriots - For Truth" party in the lower house of the Russian parliament, said that Washington had embarked on an arms race that could lead to "devastating consequences" for the United States.

"The possible use of a base in Cuba, which was recently visited by Russian ships carrying hypersonic weapons abroad, is just one of many options," Mironov said in a statement.

"Nowadays, the Cuban Missile Crisis could happen anywhere, taking into account the modern capabilities of the Russian army, air force and navy." Mironov's party is staunchly pro-Putin.

The 1962 stand-off between the Soviet Union and the United States over Moscow's stationing of nuclear weapons in Cuba is widely considered the moment when the Cold War superpowers came closest to intentional nuclear war. Moscow's invasion of Ukraine in 2022 has brought Russia's ties with the West to their lowest ebb since then.

Then-U.S. President John Kennedy discovered that Soviet leader Nikita Khrushchev had deployed nuclear missiles on Cuba after the failed Bay of Pigs invasion - a U.S.-backed attempt by Cuban exiles to overthrow Communist rule that was thwarted by Cuba - and the U.S. deployment of missiles in Italy and Turkey.

"It is worth recalling that the cause of the crisis in the 1960s was the deployment of U.S. missiles in Turkey," Mironov said.

"Now the United States is again hatching plans to deploy weapons near the borders of Russia, supply long-range weapons to the Kyiv regime, and plan and direct attacks on Russian territory."

Russian war ships including the Admiral Gorshkov frigate and the nuclear-powered submarine Kazan churned into Havana harbour last month. The submarine and a frigate carried Zircon hypersonic missiles, Kalibr cruise missiles and Onyx anti-ship missiles, Russia said.

On Oct. 27, 1962, Kennedy secretly agreed to remove all missiles in Turkey in exchange for Khrushchev removing all missiles in Cuba. Although the crisis was defused it became a symbol of the perils of superpower rivalry in the Cold War. (Reporting by Guy Faulconbridge Editing by Frances Kerry)