The maker of the iPhone is looking at a potential takeover of or investment in the British-based firm, which could be valued at up to £1.5 billion (1.7 billion euros, $1.9 billion), the newspaper said, citing three unnamed sources.
A spokesman for Apple said it would not comment on “rumours or speculation”, while McLaren did not immediately respond to a request for comment.
If such a deal went ahead, it could be seen as part of a bid by Apple to challenge Tesla, the US luxury electric carmaker founded by billionaire entrepreneur Elon Musk.
Shares in Apple were down 0.48 percent since opening in New York, standing at $113.03.
“Owning McLaren would be Apple waving a big flag over its intention to enter the car industry,” said Jasper Lawler, a market analyst with CMC Markets in London.
“As a high-end technology company, Apple appears to want to be at the high, cutting-edge end of the car industry too.”
He added: “An Apple acquisition of Tesla now looks unlikely and Tesla now faces a big new competitor.”
Apple fuelled speculation about its ambitions in the auto sector this year with a billion-dollar investment in Chinese ride-sharing company Didi Chuxing.
The New York Times reported earlier this month that Apple was rethinking its strategy, mothballing parts of its Titan self-driving cars project.
Google’s parent company Alphabet and San Francisco-based Uber have both been working on getting self-driving cars on public roads. Uber began testing self-driving taxis last week in Pittsburgh, Pennsylvania.
McLaren, which also runs a successful Formula One team, is based in Woking, southern England.