The Kremlin stunned Mr Gore and his White House entourage by declaring at the last minute that Mr Yeltsin would not see the Vice-President until today because he had decided to take a holiday.
Mr Yeltsin, 65, who has a history of heart trouble, disappeared from public view on 26 June, resurfacing only in pre-recorded television appearances, despite being re-elected in momentous style on 3 July.
The line from official sources has varied. The President has been suffering from either a cold or a sore throat, or was simply in need of a rest after his gruelling election campaign.
Mr Gore was clearly flabbergasted by the postponement, which broke all protocol rules and was announced after a large group of US officials and reporters had already arrived at the Kremlin. He may also have been shocked by the casual way in which Mr Yeltsin's aides treated the extraordinary affair.
Sergei Medvedev, his press secretary, was quoted by the Interfax news agency as saying: "It is the most convenient time to rest and restore his health after a tense election campaign, more so because the weather in the Moscow region is good for this now."
This explanation seemed almost flippant in the light of the treatment of Mr Gore, and it gave rise to speculation that, in avoiding the Vice-President for a day, Mr Yeltsin was expressing disapproval at some aspects of US policy towards Russia.
It is believed that Mr Gore intends to raise two sensitive issues with Mr Yeltsin: the recent increase in Russian military campaigning in Chechnya, and Nato's determination to incorporate new members from central and eastern Europe.
However, from the US viewpoint, the meeting was always intended to be a friendly one, and Mr Gore is still expected to congratulate Mr Yeltsin on his re-election to a four-year term. Thus Mr Yeltsin's health and caprice remain the more likely reasons for the postponement.
The Kremlin switched the venue of today's meeting to Barvikha, the village and health care centre outside Moscow where Mr Yeltsin recuperated last year from his two heart attacks.
It was at Barvikha that Mr Yeltsin cast his ballot two weeks ago, rather than make a public appearance at his normal polling station in western Moscow. Mr Medvedev said the President had not had a full medical check- up since late last year and even refused in recent weeks to have his blood pressure taken. "It is very difficult for the doctors," he said.
Mr Yeltsin's ability to take important decisions appears unaffected, as was demonstrated in a statement issued by his press office fewer than six hours after the postponement of the Gore meeting. It said that Mr Yeltsin had sacked a prominent hardliner, Nikolai Yegorov, as head of his personal staff and replaced him with Anatoly Chubais, a leading reformer and the architect of Russia's huge privatisation programme.
Mr Yegorov's dismissal completes a rout of hardliners that began in mid- June with the sacking of Pavel Grachev as defence minister, Mikhail Barsukov as head of the Federal Security Service (ex-KGB), Oleg Soskovets as a first deputy prime minister representing the military-industrial complex, and Alexander Korzhakov as Mr Yeltsin's Rasputin-like head of presidential security.
The complexion of Mr Yeltsin's administration, which had taken on an increasingly conservative hue as the election approached, is reverting now to a more reformist colour following the victory over the Communist leader, Gennady Zyuganov. However, much power is concentrated in the hands of the centrist Prime Minister, Viktor Chernomyrdin, and the national security supremo, Alexander Lebed, whose political opinions are unpredictable, but often illiberal.
Mr Chubais was the last out-and-out reformer in the Russian government before being dismissed for tactical reasons in January. He played a decisive role in organising Mr Yeltsin's campaign strategy and finances, and he had a hand in the sackings of the hardliners.