He made the comment in a 1971 telephone call with then-President Richard Nixon.
Mr Reagan, who was governor of California at the time, was angered that African delegates at the UN sided against the US in a vote.
Members of the Tanzanian delegation started dancing after the UN voted to recognize China and expel Taiwan.
Mr Reagan, who was a supporter of Taiwan, called the president the following day to express his apparent frustration.
He said: "To see those... monkeys from those African countries - damn them, they're still uncomfortable wearing shoes!"
Mr Nixon, who quit as president in 1974, can then be heard laughing.
The recording was unearthed by Tim Naftali, a clinical associate professor of history at New York University, who directed the Nixon Presidential Library from 2007 until 2011.
He published his findings in The Atlantic and wrote that the racist exchange was removed from the original tapes for privacy reasons.