A Downing Street spokeswoman told CNN the number could rise to 30, in line with authorities' worst fears. Dozens more were injured in the Friday attack.
Meanwhile, the possibility of a link to another terrorist attack emerged.
Saif Al-Deen Al Rezgui, 24, the gunman whom authorities say was responsible for Friday's deadly attack at the Sousse resort beachfront, trained with the people who carried out the attack on the Bardo Museum in Tunis in March, Rafik Chelly, Tunisia's secretary of state for security, told CNN.
In an online audio statement, ISIS -- which has said it's behind the beach resort attack -- claimed responsibility for the Bardo Museum attack and identified two men, Abu Zakariya al-Tunisi and Abu Anas al-Tunisi, as having carried it out. CNN could not independently verify the legitimacy of the audio claim.
On Tuesday, four of the most seriously wounded Britons arrived back in the United Kingdom, flown home on a military plane to receive treatment.
The gun rampage that killed 38 people is the most significant attack against British citizens since the London transport bombings 10 years ago
Tunisian authorities say they believe only one man, Al Rezgui, was responsible for the attack. Police shot him dead near the scene.
A man named Fathi was working on a nearby house during the attack, and watched the gunman die.
"He was waiting for the bullet to die, like he'd finished his mission," he told CNN.
Tunisia's Interior Ministry released details and images via Facebook of two more suspects -- described as "dangerous terrorists" -- it's seeking in connection with the attack.
The pair are identified as Rafeeq al-Tayary, 28, and Mohammed bin Abdullah bin Mohsen al-Shirady, a 23-year-old student.
Tunisian Interior Minister Najem Gharsalli announced a first round of arrests at a joint press conference Monday with his German, French and British counterparts in Sousse.
"We have started by arresting a first group, the important part of the network that was behind this terrorist criminal," Gharsalli said.
The Tunisian government has announced plans to arm its tourist police on beaches in an attempt to protect the fragile -- but economically important -- tourism industry.
But CNN teams in Tunisia say the additional security at tourist sites does not yet appear to be in place.
Tunisia: 11 victims still to be identified
The Tunisian Ministry of Health said that 19 British nationals had been identified as being among the dead as well as one Belgian, one Portuguese, one Russian, two German and three Irish citizens.
Eleven more victims remain to be formally identified. It's not clear why there's a disparity between the number of Britons identified by the British and Tunisian authorities.
The ministry said 37 of the 39 people injured in the attack had been discharged from the hospital after receiving medical care. The remaining two are still in the hospital in Sousse.
The UK government has declared that a minute's silence will be marked nationwide Friday, a week after so many lost their lives.
Tourists come to terms with terror, grief
Some tourists chose to cut short their vacation in the wake of the bloody attack.
But others remain on the golden beaches that have helped make this stretch of Tunisia's Mediterranean coast so popular with foreigners.
British couple David and Jean Rapetti, in Tunisia to celebrate their 55th wedding anniversary, are among those who decided to stay.
"We were probably about 30 yards or 30 meters away from the gunman. Which was near enough anyhow. And I was ... frightened," David Rapetti told CNN.
They witnessed the man firing and saw three people lying on the ground, their condition unclear, before finding shelter in a small room.
Staying on in Tunisia has helped them to make sense of what happened, Rapetti said. But his hardest moment since has been seeing people in terrible emotional turmoil on the beach in front of the Hotel Riu Imperial Marhaba where the attack began.
"It wasn't us that was suffering, it was other people that were suffering -- and to come to terms with their terror and their grief has taken us a lot of time," he said.
Link to Libyan terror group?
It is unclear whether ISIS had any direct role in the attack.
The Islamist group has posted a photo of the attacker, Al Rezgui, from the town of Gaafour, about 160 kilometers (100 miles) northeast of Sousse.
A spokesman for Tunisia's Interior Ministry told CNN that Al Rezgui had a connection to a Libyan terrorist organization. However, Mohammed Ali Aroui said it was unclear whether the gunman had a link to the Ansar Al-Sharia militant group or ISIS in Libya.
The ministry spokesman said it's likely Al Rezgui had spent time in Libya.
The shooting was one of three terrorist attacks on three continents
that followed one another in frightening succession Friday, also rocking France and Kuwait.
Fresh details emerged from France, where Paris prosecutor François Molins said a man accused of beheading his boss and displaying the corpse with Islamic flags had sent images of his gruesome actions to a contact in Syria
Courage of hotel workers, guests
In Tunisia, accounts of bravery have emerged from the horror and confusion of the brief gun rampage.
Amateur video filmed during the attack showed beach and hotel workers running toward a man they knew to have gunned down hotel guests just moments earlier.
British Home Secretary Theresa May reflected on that courage in remarks as she visited the scene of the attack with fellow European interior ministers.
"I've heard of the stories, the horror stories of people who went through this, and our thoughts are with the victims and the families," May said, "but I have also heard stories of great bravery as well -- the young man who shielded his fiancée from the bullets, the staff at the hotel who protected their guests."
Meanwhile, authorities in Britain were staging a two-day counterterrorism exercise in London
on Tuesday and Wednesday.
The largest such exercise to date, it will involve hundreds of government staff and Metropolitan Police officers. It was planned long before Friday's events unfolded but will seek to test emergency workers' capacity to respond to a major attack by armed terrorists.