The video allegedly shows teacher and writer Rashid Khalidi, known for his pro-Palestinian stance, along with other speakers who likened Israel and Israelis to terrorists.
Obama, who is a friend of Khalidi, is shown giving a speech in which he mentions the frequent discussions he has had with Khalidi. He also urges the people of the Middle East to find common ground.
The LA Times said it was given the video, shot in 2003, on the condition that it did not show it to anyone else.
The paper reported the tape's existence and described the events of the dinner in April. It drew little attention until raised by right-wing bloggers earlier this month.
The McCain campaign wants the tape to be broadcast, believing it could show how Obama reacted to the anti-Israel remarks.
McCain said on US radio: "I'm not in the business of talking about media bias, but what if there was a tape with John McCain with a neo-Nazi outfit being held by some media outlet?
"I think the treatment of the issue would be slightly different."
The LA Times said it had been inundated with complaints this week after conservative commentators demanded the tape's release.
However, Doyle McManus, the LA Times Washington bureau chief, said the paper was being unfairly criticised for a routine agreement with a source.
He said: "We revealed this event. We didn't suppress it."
Obama's stated support for Israel is a crucial issue to many Jewish voters in swing states such as Florida.
The Democratic Senator appeared in a 30-minute campaign infomercial on three major US television networks last night before game five of the World Series.
Estimated to cost $1m (£640,000) each, the campaign spots aired on CBS, NBC and Fox.