The vaccine's manufacturer has insisted it is safe, saying a review of available data in more than 17 million people who have been vaccinated across the UK and EU has shown no evidence of an increased risk.
After Ireland announced on Sunday that it was suspending use of the jabs as a "precautionary step", the UK's medicines regulator said the available evidence "does not suggest the vaccine is the cause" of clots.
Dr Phil Bryan, vaccines safety lead at the Medicines and Healthcare products Regulatory Agency (MHRA), said people "should still go and get their Covid-19 vaccine when asked to do so."
Northern Ireland's department of health said the rollout there will continue, "in line with MHRA guidance".
The Republic's health minister Stephen Donnelly said use of the Oxford/AstraZeneca jab was being temporarily suspended "based on new information from Norway".
The decision followed reports of serious clotting in adults in Norway which left four people in hospital.
The Netherlands also said on Sunday that it was suspending use of the vaccinations as a precaution for two weeks.
Several other European countries have already temporarily suspended use of the jabs.
But the European Medicines Agency (EMA) has said the vaccine's benefits continue to outweigh its risks and that jabs can keep being administered while it carries out a review into any incidents of blood clots - noting last week that there had only been 30 cases reported among almost five million people jabbed in the European Economic Area.
The number of cases of blood clots reported is lower than the hundreds of cases that would be expected among the general population, AstraZeneca's chief medical officer Ann Taylor said.
The pharmaceutical giant said its review had found no evidence of an increased risk of pulmonary embolism, deep vein thrombosis (DVT) or thrombocytopenia, in any defined age group, gender, batch or in any particular country.
Dr Taylor said: "Around 17 million people in the EU and UK have now received our vaccine, and the number of cases of blood clots reported in this group is lower than the hundreds of cases that would be expected among the general population.
"The nature of the pandemic has led to increased attention in individual cases and we are going beyond the standard practices for safety monitoring of licensed medicines in reporting vaccine events, to ensure public safety."
Dr Bryan said: "We are aware of the action in Ireland.
"We are closely reviewing reports but given the large number of doses administered, and the frequency at which blood clots can occur naturally, the evidence available does not suggest the vaccine is the cause.
"People should still go and get their Covid-19 vaccine when asked to do so."
The latest developments came as charities in the UK wrote an open letter to encourage people with underlying health conditions to come forward for a vaccine.
Meanwhile, MPs have called on the Government to stop "moving the goalposts" in decisions on coronavirus restrictions, urging it to publish data thresholds for its road map out of lockdown.
A report by the Commons Public Administration and Constitutional Affairs Committee said the framework for lockdown and tiering decisions changed "repeatedly", with decisions not always appearing to reflect new information.
The committee warned that this had led to confusion and mistrust among the public - despite trust being a "crucial factor" in the success of the response to the crisis.
The UK's national statistician said he has "no doubt" that there will be a further wave of Covid-19 infections in the autumn.
Professor Sir Ian Diamond, head of the Office for National Statistics (ONS), said while people need to understand how the data is moving forward and look at the impact of the "wonderful" vaccine rollout, the virus "isn't going to go away".
He told BBC's The Andrew Marr Show: "And I have no doubt that in the autumn there will be a further wave of infections."
Professor Anthony Harnden, deputy chairman of the Joint Committee on Vaccination and Immunisation, said that a booster jab could be rolled out to the nation's vulnerable and elderly in late summer in preparation for a third wave.
He told The BMJ's Talk Evidence podcast: "We certainly don't want to see a winter like we've seen this winter - and if we've got new variants circulating and we've got dropping levels of immunity due to the vaccination, then that becomes an imperative to do a booster."