Media reports said Kim gave birth to a baby girl. Messages of congratulations flooded in on Twitter.
World number two Murray, 28, joins top-ranked Novak Djokovic, 17-times grand slam champion Roger Federer and reigning French Open champion Stanislas Wawrinka, who all juggle top-flight tennis with parental duties.
Federer has fathered two sets of twins.
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While Murray has yet to officially announce the birth, his paternal grandmother Ellen Murray was quoted in the Daily Telegraph on Tuesday, saying: "Andy phoned yesterday and said 'congratulations, you're a great granny'".
"He told me the baby was 8lb 10oz when she was born. It's wonderful news. We watched him in the Australian Open final and I think he just wanted to come home."
Murray, who won Wimbledon in 2013, was beaten by Djokovic in the Australian Open final nine days ago before catching the first flight home.
(Reporting by Martyn Herman; editing by Katharine Houreld)
