The 28-year-old has played Kyrgios twice in Grand Slams this year, winning in straight sets at both the Australian and French Opens.
World number 37 Kyrgios, 20, was fined $10,000 this month for an "insulting remark" to Stan Wawrinka.
Top seed Novak Djokovic takes on Joao Souza of Brazil in New York.
Women's defending champion Serena Williams, who is aiming to complete a calendar Grand Slam, will play Russian Vitalia Diatchenko, while men's defending champion Marin Cilic will face a qualifier.
Murray could face French Open champion Wawrinka in the quarter-finals and second seed Roger Federer in the semis.
 
It is arguably the toughest possible first-round draw, it is certainly the most high-profile - Kyrgios will bring plenty of baggage on court against Murray, along with his powerful game.
The Australian, just outside the seedings, is playing under a suspended 28-day ban and $25,000 (£16,200) fine as his behaviour is closely monitored following the comments to Wawrinka.
On-court microphones picked up Kyrgios making a lewd suggestion about the Swiss player's girlfriend.
The incident made global headlines and has raised the Kyrgios profile to such an extent that it is likely his contest with Murray will be a night match.
The Australian's game, when firing on all cylinders, is good enough to have already seen off the likes of Federer and Nadal.
"Kyrgios has had a lot of trouble against Murray, who's able to defend against his game really well," said Darren Cahill, former coach to Andre Agassi, at Thursday's draw.
To add extra spice, if it were needed, Murray and Kyrgios are likely to meet again straight after the US Open when Britain face Australia in the Davis Cup semi-finals in Glasgow.
 

The main interest in the men's draw surrounded which side Murray would be on, and where eighth seed Rafael Nadal would land.
The Spaniard's slip down the rankings means he now lurks as a potential Grand Slam quarter-final opponent for his established rivals, this time Djokovic.
Nadal will need to get past Croatian teenager Borna Coric in the first round, with the 18-year-old having won their only previous meeting.
In-form second seed Federer plays world number 33 Leonardo Mayer in the first round, with Tomas Berdych a potential quarter-final opponent and Murray or Wawrinka in the semis.
World number one Djokovic remains the favourite, despite defeats by Murray and Federer in recent weeks.

Williams has won her last 28 matches at the tennis majors - she needs seven more to make history as only the fourth woman to complete the calendar Grand Slam.
The American, 34, already holds all four major titles stretching back to last year's US Open, has 21 in total and has won the last three US Opens.
Her last defeat at a Slam came against Alize Cornet at Wimbledon 2014.
Sloane Stephens, Madison Keys, her sister Venus and Maria Sharapova are potentially standing in the way this time, with Simona Halep seeded to be waiting in the final.
"I don't feel the pressure right now," Williams insisted. "I'm really just here to perform and do the best I can."