New Fine Gael leader Leo Varadkar has indicated that he intends to establish a Cabinet level committee on national security similar to the Cobra committee in the UK within the first 50 days of him becoming Taoiseach.
In a statement, a spokesperson said that although Ireland is not at high risk of a terrorist attack, it is important to be prepared for every eventuality.
The new committee will allow greater ministerial involvement in preparing for and managing major security threats, and more extensive cross-departmental cooperation on these issues.
The UK's Cobra committee deals with major crises such as terrorism.
President Michael D Higgins joined a host of world leaders in condemning last night's attack on London in which seven people were killed and more than 40 wounded.
Mr Higgins said: "This latest terrible attack in London will be rightly condemned by all those who believe in the rights of our public to live their lives in the public space.
"To our neighbours in the United Kingdom I send the sympathy and support of the Irish people, and in particular the heartfelt sympathy to the families of those who have lost their lives, and our thoughts and wishes for such recovery as is possible to the injured and their families."
US President Donald Trump spoke to Mrs May to offer his condolences for the "brutal" terror attack, the White House said.
He also tweeted condolences and support.
Mr Trump wrote: "Whatever the United States can do to help out in London and the U.K, we will be there - WE ARE WITH YOU. GOD BLESS!"
Mr Trump later issued a number of tweets on the attack, one of which apparently questioned London Mayor Sadiq Khan who earlier urged Londoners not to be alarmed to see a higher police presence on the streets following the attack.
Taoiseach Enda Kenny said in a tweet that "we stand close with our neighbours again tonight".
Terrible news coming from London - we stand with our close neighbours again tonight. @IrelandEmbGB monitoring the situation.
— Enda Kenny (@EndaKennyTD) June 3, 2017
Minister for Foreign Affairs Charlie Flanagan said we are running out of words of condemnation and described what happened in London as an attack on our fundamental values, our freedoms and our ability to enjoy life by people who have no other objective than to disrupt life and take life in an arbitrary and vicious manner.
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Pope Francis used his address in St Peter's square to urge believers to pray for the victims of the deadly attack in London.
"May the holy Spirit bring peace to the whole world, cure it of the plagues of war and terrorism, which even last night in London has struck innocent civilians," the pontiff said.
"Let us pray for the victims and those close to them."
German Chancellor Angela Merkel said that Germany was "resolutely at Britain's side".
"Today we are united across borders in horror and mourning, but also in determination," Ms Merkel said in a statement.
"For Germany, I reiterate that in the fight against all forms of terrorism, we are resolutely at Britain's side."
French President Emmanuel Macron said: "France is more than ever at the side of the United Kingdom in response to this new tragedy.
"My thoughts are with victims and their loved ones."
Two French people were injured in the attack, one seriously.
The Elysee Palace said in a separate statement that France was doing everything it could to help the casualties, adding that security would be tightened for expatriates voting in London today in the upcoming French parliamentary election.
"The attack which struck London... is a new abominable, cowardly attack on our free societies," the presidency said.
Canadian Prime Minister Justin Trudeau said Canadians "stand united" with London.
"Awful news from London tonight. We're monitoring the situation," Trudeau said on Twitter.
During a reception with parliamentary press, Mr Trudeau added: "Canadians stand united in sending our love and support to our friends in London."
European Council President Donald Tusk said: "My heart and mind are in London after another cowardly attack. Europe stands by UK in fight against terrorism."
European Commission President Jean-Claude Juncker said: "Following latest London incidents with horror. Thoughts and prayers are with the victims and their families. Please stay safe."
President of the European Parliament Antonio Tajani said: "We mourn in solidarity with the victims and the families of heinous London Bridge attack.These acts must be stopped. #UnitedAgainstTerrorism."
Spanish premier Mariano Rajoy offered "solidarity and support" and Lithuania's President Dalia Grybauskaite said that the "London spirit" was "stronger than fear".
Belgian PM Charles Michel said: "A new tragedy in London, we all stand with UK."
Australian Prime Minister Malcolm Turnbull said two of its citizens have been affected by the attack.
"This attack is yet another cruel example of the new reality in which we live, the ever-present threat of murderous Islamist terrorists, intent to harm our communities, our way of life and the freedoms we hold dear," Mr Turnbull said.
Russian President Vladimir Putin condemned the London "terrorist attack" and sent "profound condolences" to the British people, the Kremlin said.
"Putin expresses his profound condolences for the British people and condemns the terrorist attack carried out hours ago in London," Kremlin spokesman Dmitry Peskov told Russian news agencies.
