In an official response to the petition, which was launched last month and, at the time of writing, has received 16,957 signatures, the Government said it has no plan to introduce a visa route for recruitment at or near the minimum wage with relatively short training.
The Government said: "Our policies are to enable the legitimate movement of people and goods to support economic prosperity.
"It is our vision that a global and streamlined points-based immigration system and a dynamic and digitised border, underpinned by data, delivers for the UK as a global trading nation. The points-based immigration system will affect talent to the UK, irrespective of nationality."
The petition, created by Thiago Luz Togni, general manager of the soon-to-open Temper in Shoreditch, called on the Government to create a special visa for people from the EU countries to come to the UK to work in the hospitality industry for a period of up to two years to help the industry cope with the severe shortage of staff.
It would be similar to the Seasonal Work visa for horticulture workers, with the petition noting that other countries do have visas such as this to support the hospitality sector.
The Government said it recognises the ending of free movement is a significant change for businesses, and that it believes immigration must be considered alongside investment in, and development of, the UK’s domestic labour force, rather than as an alternative to it.
"This includes areas, like those in the hospitality industry, where training requirements are relatively short and should be easily accessed by the resident population. It is also worth noting similar recruitment issues for hospitality businesses are being reported in countries within the European Union, further indicating immigration is not the go-to solution for them."
It added that it continues to engage with sectors across the economy as they adapt to changes in the UK labour market.
The petition will run until 18 February 2023. If it reaches 100,000 signatures, the petition will be considered for debate in Parliament.
Job vacancies in the hospitality remain at a record high, according to data published last month by the Office of National Statistics (ONS). The ONS says that there were 173,000 vacancies recorded in the accommodation and foodservice between May and July 2022.