The world number two is also due to meet Wawrinka in the semi-finals while Australian Nick Kyrgios is a potential last-16 opponent.
Third seed Roger Federer, who demolished Murray in the last four 12 months ago, is in the same half as Novak Djokovic, meaning the Scot cannot meet either until the final.
It represents a relatively kind draw for Murray, who cruised past Wawrinka in the French Open semi-finals last month and is yet to lose to Kyrgios in four competitive meetings.
Richard Gasquet, a potential foe in the quarter-finals, has also lost his last six successive matches against the British number one.
Murray’s opener against Broady continues his recent spurt of all-British clashes. The 2013 Wimbledon champion had not faced a compatriot at tour level in almost 10 years at the start of the month but after playing two at Queen’s, this will be his third in as many weeks.
There were more fiendish propositions for Britain’s other male contenders.
James Ward has the daunting prospect of facing Djokovic, the reigning champion and current holder of all four grand slam titles, while British number two Aljaz Bedene is up against Gasquet, seeded seventh, in round one.
Kyle Edmund, the talented 21-year-old from Yorkshire, has a tough opener against Frenchman Adrian Mannarino and if he wins will face the winner of Djokovic and Ward.
Alex Ward plays Belgium’s world number 11 David Goffin, Brydan Klein is tasked with overcoming French grass-court specialist Nicolas Mahut and Dan Evans was drawn against Germany’s Jan-Lennard Struff.
