The Tunisian Interior Minister has said authorities have arrested a group of suspects associated with the man who killed 39 people in the coastal town of Sousse on Friday.
Three Irish people and at least 18 British nationals were among those who died.
The number of British nationals killed is expected to rise to "around 30", a spokeswoman for British Prime Minister David Cameron said.
Interior Minister Najem Gharsalli said officials were still verifying whether the attacker had been trained in neighbouring Libya in jihadist camps.
An Irish couple in their mid-50s, named as Laurence (Larry) and Martina Hayes from Athlone, Co Westmeath, are confirmed to have died in the attack, which took place on Friday when a lone gunman opened fire on a beach.
The couple have one daughter.
Mr Hayes was a school bus inspector with Bus Éireann and had worked at the company for 20 years, beginning as a bus driver in Galway.
The third Irish victim was Lorna Carty, a mother-of-two from Robinstown in Co Meath.
She was on holiday in the popular beach resort of Sousse with her husband Declan when the attack happened.
Footage has emerged of the gunman making his way down the beach during the attack.
Meanwhile, a couple who were staying at the Hotel Imperial Marhaba when the attack happened have been describing their experience.
Speaking to RTÉ's News at One, Anthony Tunstead said: "When it did calm down, I, actually, ventured out on to the landing; scouted around - when this guy appeared from nowhere.
"He had a kalashnikov. He didn't make eye contact with me. If he had seen me, I was definitely gone," Mr Tunstead said.
"But, because he was busy and looking for a way out ... that's why he didn't have eye contact with me.
"And he was the guy - and he was roaring Allah, or whatever, in Arabic and then went out of view.
"But, as I say, I backed off and went back to the room. And, I ran in and told them, 'Jesus, I'm after seeing him'," Mr Tunstead said.
People have been laying flowers on the beach where the attack happened.
Book of condolence for Irish victims
A book of condolence to the families of Lorna Carty and Laurence and Martina Hayes has opened at County Hall in Navan, Co Meath.
The book was opened as an expression of sympathy to the families of the deceased, "who tragically lost their lives while holidaying in Tunisia," according to a spokesperson for the county council.
Tunisian student Saif Rezgui opened fire on the beach of the hotel before making his way to the pool and hotel before being shot dead by police.
Tunisian authorities said Rezgui was not on any watchlist of known potential militants.
But one source said he appeared to have been radicalised over the last six months by Islamist militant recruiters.
The self-styled Islamic State group later claimed responsibility for the attack.
It is reported that at least 600 police and staff from the United Kingdom will form part of an international investigation.
Earlier, Mr Cameron called for a fightback against extremism.
In the Daily Telegraph, he called for "a response at home and abroad".
"To our shock and grief we must add another word: resolve. Unshakeable resolve. We will stand up for our way of life.
"We must be stronger at standing up for our values - of peace, democracy, tolerance, freedom," Mr Cameron wrote.
He called for efforts against online propaganda by groups like IS but also said there was a need to strengthen political institutions abroad.
"We must also deal with it at its source, in places like Syria, Iraq and Libya, from where ISIL is peddling and plotting its death cult," Mr Cameron wrote, using another term for IS.
Carrying out such an attack during the holy month of Ramadan was an "insult to all Muslims worldwide", Mr Cameron wrote.
"This is not the war between Islam and the West that ISIL want people (to) believe. It's between the extremists who want hatred to flourish and the rest of the world who want freedom to prosper."
Mr Cameron described the attack as another example of the "evil" seen in attacks in Iraq, Kenya, magazine offices in Paris and schools in Pakistan.
Flags were to be flown at half mast over Mr Cameron's Downing Street office today in sympathy with the victims and their families.
It was the worst loss of British life in a terror attack since 52 people died in suicide bombings on the London transport system on 7 July 2005.
