The president held back from publicly endorsing his former prime minister, who this weekend delivered Moscow's second political shock within a week by declaring that he intends to run for the presidency in 2000.
Although Mr Chernomyrdin,, dismissed by Mr Yeltsin last Monday, claimed he had his former boss's blessing to enter the race, Kremlin aides were working hard behind the scenes to emphasise that Mr Yeltsin had yet to choose his favoured candidate. Mr Yeltsin had "not determined his position" on the ex-premier's decision to run, a senior source told the Interfax news agency. Nor, the source said, had the 67-year-old president made his mind up about a third term.
A third term would seem improbable, given Mr Yeltsin's erratic health and behaviour, and his advancing years. But he is famously unpredictable, and enjoys applying egg to the faces of his doctors and critics. He also cares about his place in history as the man who oversaw the birth (albeit still incomplete) of a new Russia from the ashes of the Soviet Union. Health allowing, if he and his inner circle - some of whom have fortunes at stake - conclude that no other acceptable candidate has a chance of victory, he may yet seek to stay on.
Mr Chernomyrdin's declaration came as a surprise because, with more than two years to go, it is remarkably early in the game. The president's cool response is almost certainly also because he wants to see how his former colleague fares before committing himself.
Conventional wisdom has it that the burly former prime minister has about as much chance of victory as Mr Yeltsin has of winning Wimbledon. He is deemed to have the charisma of a tea-towel - and one soiled, to boot, by five years' service in an unpopular government.
But Russian politics have little to do with conventionality. What matters far more than the former prime minister's dreary persona is the support of the small elite who control much of the media, and whose private fortunes helped propel an unpopular Mr Yeltsin back into office in 1996.
Yesterday one of the leading king-makers, Boris Berezovsky, an oil and media tycoon, welcomed Mr Chernomyrdin's move, saying he viewed it "absolutely positively". The Kremlin's backing, even if covert, can help bamboozle the more pliable regions into lining up behind a candidate.
But equally important is the strength of the field. One likely combatant is the mayor of Moscow, Yuri Luzhkov. Like Mr Chernomyrdin, who has the backing of the mighty energy lobby, the mayor has plenty of money, powerful connections, and a strong foothold in the media. Unlike the ex- prime minister, he is a sparkling performer, no democrat, but with a convincing popular touch. To the consternation of some in the West, his market credentials are decidedly shaky. If he decides to run, it will deepen the risk of splitting the anti-Communist vote - a prospect that will be preoccupying the king-makers. What fabulous price could Mr Luzhkov now ask in order to stay out?
Mr Chernomyrdin's 35-year-old replacement, Sergei Kiriyenko, is facing a more pressing issue. The Communists, who dominate the State Duma (lower house), intend to oppose confirming him in his post. A long bartering process between parliament and the Kremlin is about to begin.