Islamic extremist Seifeddine Rezgui slaughtered a total of 38 people, including 30 Britons and three Irish citizens, at the five-star Riu Imperial Marhaba Hotel in Sousse.
A rail worker and his wife, Christopher and Sharon Bell, from Killingbeck in Leeds, were among the victims.
More than 80 people, including relatives who lost loved ones and those left with injuries, have now instructed lawyers to commence civil proceedings, law firm Irwin Mitchell said.
The company said it represents the families of 22 people who died in the attack as well as more than 50 others who suffered injuries including gunshot wounds and people hit by shrapnel from explosions.
The legal case centres on security at the hotel, what was known about previous attacks in Tunisia, and the lack of information presented to customers both at the time of booking and when the situation may have changed regarding travel advice, the lawyers said.
Judge Nicholas Loraine-Smith, the coroner conducting the inquests of the 30 Britons, ruled in February 2017 that they were unlawfully killed.
But he ruled against a finding of "neglect" by TUI, or the owners of the hotel.
The families of the dead were highly critical of security at the hotel, which only had a handful of unarmed guards on duty when Rezgui struck, armed with an AK-47 assault rifle and home-made grenades.
They also believed that TUI did not do enough to warn holidaymakers before they booked about the dangers in Tunisia, which had suffered a fatal terrorist attack in the capital, Tunis, just three months earlier.
This included making them aware of official Foreign Office travel advice which warned of a high threat of terrorism.
Kylie Hutchison, a specialist personal injury lawyer at Irwin Mitchell, representing the families affected, said: "On behalf of our clients who lost members of their families and those who suffered injuries in this terrible incident, we have now served formal civil proceedings on TUI, claiming damages.
"The damages claimed will help compensate them for their suffering, their financial losses, and help survivors meet the costs of specialist treatments and therapies to aid their recoveries.
"The level of terrorist threat in Tunisia had been escalating for some time prior to June 2015.
"This included a failed suicide bomb attempt outside a beach hotel in Sousse in October 2013 and an attack at the Bardo museum in Tunis in March 2015 in which 22 people were killed.
"Despite this, TUI, the tour operator who organised the holidays and was responsible for our clients' safety, did not audit the adequacy of security at the hotel or take appropriate precautions to keep our clients safe from an attack.
"Nor did they inform our clients of the level of threat of terrorism which many of the holidaymakers say would have changed their mind about holidaying in Tunisia at the time."
Mat James, 33, from Pontypridd, South Wales, was shot multiple times while protecting his girlfriend during the attack in June 2015.
He has since had multiple operations on his leg and is still suffering from problems related to the injury.
He said: "The horrible attack was obviously life-changing for so many people. Even now, a few years on, my injuries are still affecting me.
"We can never forget what happened and I'm lucky to be alive, but hopefully, by taking legal action, everyone involved can get the help and support we need to aid our recoveries as much as possible."
A TUI UK spokesman said: "We remain truly saddened by what happened on that fateful day in Sousse in June 2015 when 30 of our customers lost their lives in a terrorist attack which started on a public beach.
"Our thoughts remain with all of those who were affected by the horrific incident.
"As this is now subject to legal proceedings it would be inappropriate to comment further at this stage except to say we will fully cooperate with the judicial process."