Anyone already in Italy who recently travelled from Britain must be tested for coronavirus, the statement added.
Belgium said it would close its borders to flights and trains - including the popular Eurostar service - coming from the United Kingdom.
Both are offering to reschedule passengers for free or give full refunds.
Ireland dey expected to restrict flights and ferries from midnight (23:00 GMT) on Sunday.
France has also banned accompanied freight - such as lorries - will not be allowed to enter France from midnight tonight in France.
Mr Donnelly said the vast majority of Irish people overseas had decided not to travel due to the virus.
Johnson said Saturday that a fast-moving new variant of the virus that is 70% more transmissible than existing strains appeared to be driving the rapid spread of new infections in London and southern England.
Despite the introduction of the travel ban, there is no plan to implement similar restrictions between Northern Ireland and the Republic.
The party's TD Duncan Smith urged the Government to bring forward the Cabinet meeting scheduled for Tuesday, and to "ramp up" testing and tracing capacity over the coming days.
The British prime minister also said London and southeast England, which are currently in the highest level of a three-tier system of rules, would now be placed in a new Tier 4 level.
"We've done a good job in this country".
"If everything goes well, that we may see a circumstance whereby late spring, you know, people who are in lower-risk categories can get this vaccine but that would really require everything to go exactly on schedule", he said.
"We are very concerned about it and we'll be making an announcement today on potential travel restrictions".
"I've been talking to the Taoiseach and Minister Ryan and the Chief Medical Officer over the weekend".
He said people should not meet up and warned against being in crowds.
Belgium has one of the worst Covid-19 deaths per capita rates in the world.
That is almost double November's death toll of 5,796.
The Department also reported 764 new cases of the virus.
She also said there was no evidence it was causing a disproportionate number of hospital admissions.
The Dutch ban came into effect from 6:00 am (0500 GMT) and will last until January 1.
Public health bosses have recommended stricter measures to Government, including that pubs and restaurants be closed before the new year and greater restrictions be placed on household visits after Christmas.