The Tesla and Space X CEO posted a message on Twitter to the Russian president inviting him to join the platform, which allows users to share conversations on a variety of topics.
Tagging the Kremlin’s official presidential Twitter account, Mr Musk tweeted: “Would you like join me for a conversation on Clubhouse? [sic]”
In a follow-up tweet, the world’s richest man wrote in Russian: “It would be a great honour to talk to you.”
Mr Musk has been one of the loudest backers of the new live podcasting app.
Last month he held a much-talked about Clubhouse conversation about the recent Gamestop phenomenon with investing app Robinhood’s CEO, Vladimir Tenev.
The popularity of the much-hyped app, which has been touted as a future rival to established social media giants such as Instagram and Snapchat, has in part been driven by Mr Musk’s apparent enthusiasm for it on Twitter.
It has attracted a string of well-known users include Oprah Winfrey and Mark Zuckerberg.
But Clubhouse, which is restricted to owners of iPhones, has also drawn criticism from those opposed to its invite-only template, which means people can be blocked from listening in.
Last week Mr Musk tweeted that he had agreed to host a conversation with Kanye West. “The most entertaining outcome is the most likely,” the billionaire entrepreneur said.
Clubhouse has been downloaded more than 5 million times, according to analytics firm App Annie, and the San Francisco-based company was recently valued at $1bn – having reportedly reached the $100 million valuation when it had just 1,500 users.
The Independent has contacted the Russian government for comment.
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