The Australian ambassador to the United States, Joe Hockey, put the request in to Norman to provide Trump's contact details, the golfer said in a statement.
Norman said it was a "pleasure and an honour" to facilitate the connection between Prime Minister Malcolm Turnbull and Trump, who he called a friend.
"I have great respect for both men who have been voted in by the people of their respective countries," Norman said. "I am fortunate enough to call Mr. Trump a friend, so I was happy to put them in touch to further the incredible long-standing relationship the two countries have experienced."
Norman, who was born in Australia but spends much of his time in the U.S., was ranked the number one ranked golfer in the world for more than 300 weeks during the 1980s and 1990s.
"I am confident that Australia and the United States will continue to work together in the same close spirit that they always have," he said.
Last week, Turnbull said he had a warm and "very frank" conversation with Trump within hours of his election victory, according to The Associated Press. Turnbull described Norman as a "great advocate for strengthening the Australian-American alliance" and called him a "great Australian."
Trump tweeted Wednesday that he had "received and taken calls" from several foreign leaders, including representatives of Russia, the United Kingdom, China, Saudi Arabia and Japan.