The decision comes after Irish Health Minister, Simon Harris, called for the games, which were due to be played this weekend (29 February and 1 March) in Dublin, to be put on hold.
In Italy, 322 people are confirmed to have the Covid-19 strain of coronavirus - by far the largest outbreak of the disease in Europe - and 11 people have died.
Which matches have been put on hold?
Ireland’s men's squad were set to play Italy at the Aviva Stadium in Dublin on Saturday (7 March), before the women’s match at the Energia Park on Sunday (8 March).
The matches have been postponed until further notice.
The under-20s contest between the two nations, scheduled for 6 March, has also been put on hold.
When will the matches be rescheduled?
The Irish Rugby Football Union (IRFU) said it hoped to give a precise date for the rescheduling “in the coming days”.
On Sunday, Italian Prime Minister, Guiseppe Conte, gave regional authorities “extraordinary powers” to help contain the spread of the disease, including the ability to prevent travel between population centres.
Ten small towns in the northern region of Lombardy, and one town in Veneto, have been locked down by police.
“In the interests of national Public Health”
An IRFU statement said: "At the outset we made it clear that the IRFU was supportive of the government's need to protect public health in relation to the coronavirus.”
"We were then advised, formally, that the National Public Health Emergency team has determined that the series of matches should not proceed, in the interests of Public Health.
"We will immediately begin to work with our Six Nations partners to look at the possibility of rescheduling the matches and would hope to have an update on this in the coming days."
On Sunday, Scotland’s women’s match against Italy was postponed, and England rugby authorities confirmed that they were monitoring the situation ahead of the England v Italy match in Rome on 14 March.
And on Monday, rugby officials postponed Ospreys and Ulster’s Pro14 tours to Italy on 29 February. Ospreys were meant to face Zebre in Parma, and Ulster were set to play Benetton in Treviso.
The Italian Rugby Federation also suspended all National Championship matches for the weekend of 28 February.
World sporting events threatened by coronavirus
Rugby is not the only sport being impacted by the increasing concern caused by coronavirus.
Amid the country’s ongoing battle to contain the spread of the disease, Italian officials ordered five matches in the next round of its Serie A football competition to be played without fans present.
And in South Korea, where 977 people are infected with Covid-19, authorities have cancelled several sporting events.
Earlier today it was announced that the World Short Track Speed Skating Championships - due to be held in Seoul in March, would be cancelled.
Korean authorities also postponed the World Table Tennis Championships, scheduled for March, until the end of June.
Meanwhile, in Vietnam, where there are 16 Covid-19 infections, officials confirmed that the country’s inaugural Grand Prix will still go ahead.
Formula 1’s managing director, Ross Brawn, said: "For Vietnam, all the feedback we are getting is rather like the UK. There have been some cases but not a level that would concern us.”
The race is set to go ahead in Hanoi, despite the government announcing a 10,000-person quarantine in the commune of Son Loi, just 30 miles north-west of the capital.