KIEV, Ukraine — The tense Russian-American jockeying over the fate of Ukraine escalated on Thursday as a Kremlin official accused Washington of “crudely interfering” in the former Soviet republic, while the Obama administration blamed Moscow for spreading an intercepted private conversation between two American diplomats.

An audiotape of the conversation appeared on the Internet and opened a window into American handling of the political crisis here, as the two diplomats candidly discussed the composition of a possible new government to replace the pro-Russian cabinet of Ukraine’s president, Viktor F. Yanukovych. It also turned the tables on the Obama administration, which has been under fire lately for spying on foreign leaders.

The developments on the eve of the Winter Olympics opening in Sochi, Russia, underscored the increasingly Cold War-style contest for influence here as East and West vie for the favor of a nation of 45 million with historic ties to Moscow but a deep yearning to join the rest of Europe. The tit for tat has been going on since November, when Mr. Yanukovych spurned a trade deal with Europe and accepted a $15 billion loan from Moscow. Months of street protests have threatened his government, and American officials are now trying to broker a settlement — an effort the Kremlin seems determined to block.

The posting of the audiotape represented a striking turn in the situation. It was put anonymously on YouTube on Tuesday under a Russian headline, “Puppets of Maidan,” a reference to the square in Kiev occupied by protesters, and then posted on Twitter on Thursday by a Russian government official who called it “controversial.”