The conversations between the two have revolved around everything from geopolitics and business to personal matters, according to several of the newspaper's unnamed sources in the USA, Russia, and Europe.
The Kremlin's spokesperson, Dmitrij Peskov, tells WSJ that Putin and Musk have spoken to each other over the phone on one occasion and that the conversation centered on "space as well as current and future technologies".
Musk, who has not commented on the reports, is currently campaigning with Donald Trump in his attempt to become the next President of the USA.