Video analyst Mario Masha tested positive on Saturday and the team is in quarantine pending more test results.
Another athlete from outside the village also tested positive on Sunday.
"We have three positive cases of Covid-19 in the camp here, two players and an official," said South Africa team manager Mxolisi Sibam.
"There is daily screening which included taking temperature and saliva testing, Masha and Monyane reported high temperatures and positive saliva tests and were then taken to do the nasal test, which we all had initially taken and they unfortunately tested positive for Covid through that test. Mahlatsi is the latest player to go through the same process," added Sibam.
Monyane, a right-back with Orlando Pirates in his homeland, posted on social media when the team arrived in Tokyo.
South Africa Sevens rugby coach Neil Powell also tested positive on Sunday on arrival in Japan and is now isolating in Kagashimo, where the squad is based for a pre-Games training camp.
In total, organisers on Sunday reported 10 new cases connected to the Olympics including media, contractors and other personnel. That compares with 15 new cases on Saturday.