The state-owned channel, which promotes the armed forces but also airs general entertainment shows, published a letter to Clarkson on its website asking him to host a new motoring show and inviting him to visit Moscow to discuss the offer.
"He will reply. I cannot reply on his behalf," a spokesman for Clarkson named by the channel as Daniel Rix told the channel by telephone.
It said: "I can only imagine what an episode of Russian vyshaya peredacha (a literal translation of Top Gear into Russian) would look like."
"It could feature a Tzar in a reasonably priced car!" Rix added jokingly, referring to a celebrity feature on the show called "a star in a reasonably priced car."
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It wasn't clear whether Zvezda got the point of Rix's play on words, since its own translation of the letter was more literal, saying the show could become a "tsar among shows about reasonably priced cars in Russia."
Top Gear draws more than 350 million viewers around the world and is popular in Russia, where the show airs on state television. A licensed Russian local version ran for a while but did not prove a hit. Clarkson has visited Russia to film the programme and present roadshows for fans.