Bruce Wilkinson, from Goole, is widely reported to have been killed in the attack.
The town’s MP Andrew Percy, tweeted: “Sad to learn yesterday of the death of a constituent in Tunisia. With names now released I am sure all our thoughts are with family here.”
There was relief for Carol Mulhall, from Doncaster, who finally heard her parents Margaret and Dougie Lamb were safe.
She had shared their photos with the BBC after failing to get hold of them following the atrocity and someone at the embassy in Tunisia recognised the couple and made contact with Ms Mulhall via Facebook.
She said: “I have never felt so thankful for anything in my life.”
Theresa May suggested that British officials were facing difficulties in identifying the victims because few of the holidaymakers were carrying anything with them.
The Home Secretary said Britons gunned down in the “appalling” massacre would not have had passports or other forms of identification with them on the beach and some victims were being moved between hospitals in the country.
She told BBC One’s Andrew Marr Show: “The figure of confirmed dead at the moment remains at 15 but we are expecting that to rise.”
David Cameron warned that the public needs to be prepared for the fact that “many of those killed” in the “savage” Tunisian shooting were Britons.
He spoke as reports from Tunisia suggested that the UK death toll had risen to eight out of 39 people killed in the popular resort of Sousse.
Mr Cameron said a “full deployment team” of consular staff, police and experts from the Red Cross would arrive in the North African country today to help the victims and their families and the Government was doing “all we can to” help them.
Speaking in Downing Street, he said: “These savage terrorist attacks in Tunisia, Kuwait and France are a brutal and tragic reminder of the threat faced around the world from these evil terrorists.”
Five of the dead have so far been confirmed as Britons by Foreign Secretary Philip Hammond, as he warned that number could rise due to the popularity of the area with tourists from the UK.
Tunisian PM Habib Essid has since reportedly told a news conference that the majority of victims were British, followed by Germans, Belgians and other nationalities.
Terror group Islamic State claimed responsibility for the attack in which a young man pulled out a Kalashnikov hidden in a beach parasol and shot at sunbathers in Sousse.
SITE Intel Group, which monitors the jihadist threat, tweeted: “#ISIS claimed credit for the Tunisia hotel attack in Sousse and identified the attacker as Abu Yahya al-Qayrawani.”
• Sousse is a city on the east coast of Tunisia, about 87 miles (140km) south of the capital, Tunis.
• Around 1.2 million tourists visit Sousse every year, drawn by the hotels, sandy beaches and culture.
•Hotel complexes with 40,000 beds span from the old city to the Port El Kantaoui and the city is home to a Unesco-protected mosque, as well as a historic medina.
•Boujaafar Beach stretches from the Gulf of Hammamet several miles north to Port El Kantaoui - a purpose-built resort with dozens of hotels including the El Mouradi Palm Marina, El Mouradi Palace and Riu Imperial Marhaba.