Local authorities have quarantined all the golf course's employees who came in contact with the two men. The entire course in Hoa Vang District, the central city of Da Nang has been disinfected.
The Britons, aged 60 and 66, are among 11 foreigners so far diagnosed with the virus in Vietnam.
They all had flown to Hanoi from London on Vietnam Airlines flight VN54 on March 2 with Nguyen Hong Nhung, Hanoi’s first Covid-19 patient and the nation’s 17th. Ten are Brits and one Irishman. A Vietnamese on the same flight has also tested positive for the coronavirus.
There were 201 passengers on the flight, including 21 in the business class, where the trio sat.
On March 2 the two Britons took a flight from Hanoi to Da Nang and stayed at a hotel on Nguyen Van Linh Street. On March 4 they went to the golf club to play.
They are being treated at Da Nang Hospital.
Vietnam has reported 22 Covid-19 cases since last Friday after going 22 days without any new infection.
The Covid-19 outbreak has thus far spread to 119 countries and territories around the world, killing 4,300 people.