There had been fears that the games on March 14 between Italy and England in Rome and France and Ireland in Paris would be called off.
But representatives of the Six Unions and championship officials met in Paris and agreed that the two matches and the other games in England, Scotland and Wales over the final two rounds of the competition would go ahead.
"As it stands today, based on the latest information, all Six Nations matches currently scheduled are set to go ahead," Six Nations said in a statement.
"Six Nations is in contact with FIR and RFU regarding the possibility of relocating the Women's and U20 Italy vs England matches to another Italian location and we will make a further announcement on this in due course."
"However, the Italy vs England senior men's match in Rome is planned to go ahead as scheduled."
The organisers said they intend to complete all 15 games across the three championships but would not commit to rescheduling fixtures while the situation was being assessed.
England's fly-half George Ford (C) runs through the Ireland defence during the Six Nations international rugby union match between England and Ireland
Image credit: Getty Images
Organisers also said they were still preparing for potential changes, should the relevant governments take further action, while a new date for the postponed Ireland-Italy game has yet to be agreed.
The decision comes after French Health Minister Olivier Veran announced a temporary ban on all public gatherings with more than 5,000 people to try to curb the spread of the disease.
Italy has more than 2,000 confirmed cases of coronavirus since the contagion came to light in northern regions on February 20 and 52 people have died.
The Italian government has extended the suspension of all sporting events until March 8 in the affected areas of Lombardy, Veneto and Emilia-Romagna, although Rome is currently not affected by those suspensions.
Ireland's match against Italy in Dublin was postponed on government advice due to the potential threat posed to public health, along with the women's and under-20 matches.
After the first three rounds of the Six Nations, Ireland and England both have nine points, four behind leaders France. Italy are bottom of the standings without a win.