Murray, 30, is in the same half of the draw as 2015 champion Stan Wawrinka, who he beat in last year's semi-final.
Second seed and defending champion Novak Djokovic faces Spain's Marcel Granollers, and is in the same half as nine-time champion Rafael Nadal.
The tournament starts at Roland Garros in Paris on Sunday.
Seventh seed Konta, who is chasing a first win in the Grand Slam event, having lost in round one in the previous two years, is up against the world number 109 from Taiwan.
Defending women's champion Garbine Muguruza opens against 2010 winner Francesca Schiavone and could face a semi-final against world number one Angelique Kerber, who starts against Ekaterina Makarova.
Romania's world number four Simona Halep, who is battling to be fit after tearing ankle ligaments, starts against Slovakia's Jana Cepelova and is in the same half of the draw as Elina Svitolina, who beat Halep in the Italian Open final last weekend.
Petra Kvitova is set to announce later on Friday whether she will play in the French Open, six months after she suffered serious hand injuries when she was stabbed by an intruder in her home, with the Czech drawn against American Julia Boserup.
Konta is the only British player in the women's singles after Heather Watson lost 6-0 6-3 to the Netherlands' Richel Hogenkamp in the third round of qualifying on Friday.
The 26-year-old has played Hsieh twice before, with the Briton winning their last match in Vancouver in 2013 and Hsieh winning on grass at Eastbourne in the same year.
Last year's runner-up Murray has won his two previous meetings with world number 85 Kuznetsov - in the third round of the 2014 US Open and again on hard court in Beijing last year.
The other British men in the draw all avoided seeded players.
Kyle Edmund, ranked second in Britain and 50 in the world, takes on Portugal's Gastao Elias, who is 75 places below him.
Dan Evans plays Spanish veteran Tommy Robredo, while Aljaz Bedene starts against American Ryan Harrison.
Former British number one Greg Rusedski
Murray has played a bit too much tennis. He got shingles, and all of a sudden he was back on antibiotics leading up to French Open. Physically he is not in the right place.
When you are not winning matches, your mental side starts to affect you, so it's a combination of the two areas.
I don't expect that much from him. I think his focus will probably be on defending his Wimbledon title. It's going to be a hard one.
Halep was the one everyone was looking at as favourite but she has had an ankle injury. If Konta can get her movement on the clay right she has got a shot, the women's event is wide open.
BBC tennis correspondent Russell Fuller
Nadal, Djokovic and Dominic Thiem have all been drawn in the other half, but that does not feel especially relevant to Murray just now.
He has lost half of his clay matches so far this year and is seeded to face Juan Martin del Potro in the third round. Nadal has an awkward opening round against Benoit Paire, but otherwise appears to be in a relatively benign quarter.
Konta is also in a reasonable quarter - avoiding Halep, Kerber and defending champion Muguruza - but given her struggles on the clay, and the lack of any obvious favourite, don't read too much into that.
Kerber lost in the first round last year and has a horrible opener against the two time Grand Slam semi-finalist Ekaterina Makarova.
World number one Andy Murray will play Russia's Andrey Kuznetsov in the French Open first round while fellow Briton Johanna Konta faces Hsieh Su-wei.