Former champion Murray has never played Broady, the world number 234 who has received a wildcard to the tournament.
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Compatriot James Ward, also a wildcard, will play defending champion Novak Djokovic in his opening match.
In the women's singles, British number one Johanna Konta faces Puerto Rican Monica Puig, while Serena Williams will face a qualifier or lucky loser.
Sixteenth seed Konta, ranked 18th in the world, has met Puig once.
The Australian Open semi-finalist came from behind to win 4-6 6-4 7-6 at last year's grass-court tournament in Nottingham.
British number two Heather Watson starts against Annika Beck and could then face top seed Williams in a repeat of last year's gripping third-round match, while Laura Robson has drawn Australian Open champion Angelique Kerber.
Robson and Kerber have met three times, including at Wimbledon in 2011.
The Briton won in three sets then, but Kerber has triumphed in the other two matches, the most recent in Beijing in 2013.
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In the men's draw, Murray is in the more favourable half and would only face Djokovic or third seed Roger Federer in the final.
Djokovic and Federer are scheduled to meet in the semi-finals.
If Ward overcomes Djokovic in round one, he could play countryman Kyle Edmund in round two.
Edmund takes on Frenchman Adrian Mannarino.
The highest seeds in Murray's half of the draw are Stan Wawrinka, Kei Nishikori and Richard Gasquet.
Frenchman Gasquet, a semi-finalist last year, will play British number two Aljaz Bedene in his opening match.
Women's singles:
Naomi Broady v Elina Svitolina (17)
Katie Swan v Timea Babos
Tara Moore v Alison van Uytvanck
Men's singles:
Dan Evans v Jan Lennard Struff
Brydan Klein v Nicolas Mahut
Alex Ward v David Goffin (11)
BBC tennis correspondent Russell Fuller:
Murray won't be looking too far ahead - he was taken to five sets in his opening two matches at the French Open - but a fourth-round match with Nick Kyrgios on Centre Court could be fraught with danger.
It could be considered an advantage for Murray that Federer is in the other half of the draw with Djokovic, but the seven-time Wimbledon champion, 34, has played little tennis in recent months.
Robson's draw suggests her recent luck is showing no sign of improvement, while Konta will have her work cut out to live up to her seeding.
She could face the 2014 runner-up Eugenie Bouchard in the second round and former Australian Open runner-up Dominika Cibulkova in round three.
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Andy Murray will face fellow Briton Liam Broady in the first round of Wimbledon, which starts on Monday.