The independent travel agent is buying the shops from the Official Receiver, which was appointed when Thomas Cook collapsed last month.
Sunderland-based Hays said it planned to reopen the shops under its own brand with immediate effect.
Hays has already offered jobs to 600 Thomas Cook staff and it plans to hire more as part of its rapid expansion.
The move is a significant step for Hays, which currently has 190 shops across the UK.
John Hays, who set up the the company 40 years ago, said: "It is a game-changer for us, almost trebling the number of shops we have and doubling our workforce - and for the industry, which will get to keep some of its most talented people".
He and his wife Irene own the business which has 1,900 staff and last year had sales of £379m, reporting profits of £10m.
The price of the deal was not announced.
Irene Hays, chair of Sunderland-based Hays, said: "Thomas Cook was a much-loved brand and a pillar of the UK and the global travel industry."
We will build on the good things Thomas Cook had - not least its people - and that will put us in even better stead for the future"
More than 100 new jobs will be based at the company's Sunderland headquarters, with the rest in shops across the UK.
The company has tweeted, urging former Thomas Cook staff to apply.
The company is telling would-be recruits that as a result of the deal "we can begin to re-open Thomas Cook stores, in their new Hays livery, from today."
"We aim to open as many as we logistically can as soon as possible," it said.
When Thomas Cook collapsed, it put 22,000 jobs at risk worldwide, including 9,000 in the UK.
It also sparked the biggest ever peacetime repatriation by the Civil Aviation Authority (CAA) to bring more than 150,000 British holidaymakers back to the UK. The last flight to repatriate Thomas Cook customers landed at Manchester Airport on Monday.
Business Secretary Andrea Leadsom said she hoped the deal "will provide significant re-employment opportunities for former Thomas Cook employees, alongside the advice and support we will continue to provide to help people find a new job as quickly as possible."
'Deal with landlords'
The Transport Salaried Staffs' Association (TSSA) union, which had members in Thomas Cook shops around the UK and in its head office in Peterborough, welcomed the move.
Manuel Cortes, TSSA's general secretary, said it offered "real hope of reemployment to former Thomas Cook retail staff, many of whom are our members."
"We will now move swiftly to gather more details about the deal and are seeking an urgent meeting with Hays about the part our union can play in supporting what will be a significant expansion for them across our High Streets," he added.
David Chapman, the Official Receiver, described the sale to Hays Travel as an "important step" in unravelling the finances of the 178-year-old chain.
The business is thought to have a licence for six months to occupy Thomas Cook stores, giving Hays time to strike new deals with landlords.
What is Hays' plan?
Ian Bell, head of travel and tourism at accountancy firm RSM, said it was a "shrewd move" for Hays, but would also represent a quadrupling of its travel agency stores at a time when customers are increasingly booking holidays online.
"Much may depend on the deals that Hays can strike with its new High Street landlords," said Mr Bell.
As Hays bought the stores from the administrators, it means it will have bought them at a lower price than if Thomas Cook was still trading.
Julie Palmer, partner at Begbies Traynor, said that in waiting, Hays had got a "best price" for the stores.
However, she questioned the logic for the transaction at a time when customers are turning away from High Street travel agents.
"You have to wonder what Hays' plan is and how they can make it a success.
"Has the travel firm been gung-ho in trying to secure a cheap deal without assessing the viability of taking on these stores?," she asked.
All 555 Thomas Cook shops are to be bought by rival Hays Travel in a move that could save up to 2,500 jobs.