Forty-one Britons detained after a raid on a Gaza-bound aid flotilla are expected to be deported "very quickly", the foreign secretary has said.
At least nine people were killed when commandos stormed a boat, but no Britons are thought to be among them.
William Hague said Britons held in southern Israel's Beersheba prison were given consular assistance. Another Briton has already been deported.
The UN Security Council has called for an immediate and impartial inquiry.
Mr Hague said the numbers were not certain as some people had destroyed their passports, but it is thought among those detained were 30 British nationals and 11 dual nationals, one of whom has received medical attention for injuries.
"They are all in the process of being visited and Israel is giving us the appropriate consular access as, of course, they should," he said.
Mr Hague said the incident should "underline, not undermine" the need for "a durable solution"
A British consular official told the BBC he had visited those detained in the prison and collected details of family and friends who they wanted to have contacted.
He said the prison was newly-built and in a pristine condition.
A Foreign Office spokeswoman said consular staff had visited 29 of the Britons with others expected to be seen on Wednesday.
She said: "There have been no complaints about their treatment."
'Cross our fingers'
Families are facing an anxious wait to hear whether relatives travelling on board the flotilla are safe.
Dr Khalid El-Awaisi said he was worried that his brother Ali, from Dundee, may have been injured.
Winnie Chambers is also waiting for contact from her sister, Theresa McDermott, from Edinburgh.
"The normal thing that happens is all communications are jammed and any communication devices are removed from the passengers," Ms Chambers told the BBC.
"We just cross our fingers and hope."
And Eleanor Lamb, of County Donegal in the Irish Republic, said she had been unable to get in touch with her son, Fiachra O'Luan, who was among the activists.
"He's courageous, and some might say foolish, but I just hope he will be OK," she told the BBC, through tears.
Deputy Prime Minister Nick Clegg called for the "unjustified and untenable" blockade of Gaza - intended to stop arms smuggling - to be lifted.
"What is going on in Gaza is a humanitarian catastrophe," he said.
"While of course Israel has every right to defend itself and its citizens from attack, we must now move towards lifting the blockade from Gaza as soon as possible."
His remarks echo those of David Cameron, who told the Israeli prime minister on Monday that he "deplored the heavy loss of life".
A Downing Street spokesman said: "He reiterated the UK's strong commitment to Israel's security, but urged Israel to respond constructively to legitimate criticism of its actions, and to do everything possible to avoid a repeat of this unacceptable situation."
Mr Cameron also stressed the need to urgently lift the three-year blockade and to allow full access for humanitarian aid, the spokesman said.
Israeli ambassador in London Ron Prosor said the military operation was "not successful" and the loss of life was tragic.
But he said the passengers had tried to provoke them despite repeated warnings against breaking the blockade.
"The people on board the ships behaved appallingly. They used iron bars, metal pipes, knives with huge blades," he said.
Britain's ambassador to the UN from 1998 to 2003, Sir Jeremy Greenstock, said the time had come for serious international action to end Israel's blockade and what he called "the virtual starvation of Gaza".
On Monday, protests were held in London, Bristol and Manchester, where campaigners targeted a BBC building, smashing doors and placing a Palestinian flag on the roof.
The ship, carrying 10,000 tons of aid, including cement and building materials that Israel bans from Gaza, left the coast of Cyprus on Sunday.
It was due to arrive in Gaza on Monday, but Israelis boarded it overnight in international waters and clashed with hundreds of passengers, who were mostly Turkish.
The passenger ferry, Mavi Marmara, was one of three provided by Insani Yardim Vakfi (IHH), a Turkish aid organisation with links to the Turkish government.
The ships have now been escorted to the Israeli port of Ashdod.
Hundreds of activists are understood to have been detained in southern Israel, with Britons among them.
Shadow foreign secretary David Miliband said: "The particular circumstances of this case will have to be brought to the surface but the defence of a failed policy that is benefiting no-one - not the people of Gaza, not the people of Israel either - I think is badly flawed."
Former Prime Minister Tony Blair, who is the Quartet Representative in the Middle East, expressed his "deep regret and shock at the tragic loss of life".
In 2007, Israel and Egypt tightened a blockade of Gaza after the Islamist movement Hamas took power there.
Israel says it allows about 15,000 tonnes of humanitarian aid into Gaza every week, but the United Nations says that is less than a quarter of what is needed.