MOSCOW—In his first public comments on the political scandal that has engulfed the Trump administration, Russian President Vladimir Putin said he wouldn’t oppose his conversations with President Trump being made public, a reversal of the Kremlin’s position opposing the publication of the two leaders’ talks.

The Kremlin has until now voiced opposition to the publication of the conversations between Messrs. Trump and Putin. Exchanges between the two leaders have generated intense interest in light of evidence that Russia interfered...