Ambassador Vitaly Churkin received the letter last week, the UN mission said on Monday in a statement, which stressed that Russia did not finance any foreign political activity.
Brian Rogers, a spokesman for McCain's election campaign, told the BBC the campaign had not itself received any complaints from the mission and that "it sounds like they're having a little fun at our expense."
"It's just an error," he was quoted as saying.
It is illegal for presidential campaigns to accept donations from foreign nationals.
Senator McCain, the Republican presidential candidate, has been a harsh critic of Russia, most recently over the conflict with Georgia following Tbilisi's attack on its breakaway republic of South Ossetia.
On August 12, as Russian troops were concluding their operation in Georgia, McCain said he had told Georgian President Mikheil Saakashvili: " I know I speak for every American when I say to him, 'Today, we are all Georgians.'''
Last year he said the G8 should exclude Russia, citing "diminishing political freedoms, a leadership dominated by a clique of former intelligence officers, [and] efforts to bully democratic neighbors."
The Russian diplomats may have taken the letter as an opportunity for a gentle push back.
But the McCain campaign has been able to laugh it off. According to the Washington Post, when asked whether there were any other ambassadors that they planned to solicit, Rogers laughed and said, "I hope not. Maybe we'll find out tomorrow."
Ruslan Bakhtin, press secretary of the Russian mission, said the letter was dated September 29 and carried McCain's signature. It was addressed to Vitaly Churkin, Russia's envoy to the United Nations, and arrived on October 16.
The ambassador's title was not included in the letter, which Bakhtin said was an indication it had been sent in error.
Enclosed was a request for a donation of up to $5,000 to McCain's election campaign to be returned with a check or permission to withdraw the money from the donor's credit card until October 24.
McCain trails his Democratic rival, Senator Barack Obama, in the polls, and in the money stakes. Obama announced on Sunday that his campaign had raised a record $150 million in September and he has massively outspent McCain, who cannot solicit private donations after accepting the $84 million available in federal funding.
