Under the rules, Hong Kong residents who hold a British National Overseas (BNO) passport will be allowed to live in the United Kingdom for five years and then apply for "settled status" and citizenship.
About half of the 5,000 applications received were from Hongkongers who were already in Britain, The Times reported, citing unidentified sources.
Some 5.4 million Hongkongers could be ultimately be eligible for British citizenship under the scheme.
Those people had already been offered temporary settlement in the United Kingdom after fleeing China's security crackdown while waiting for the visa change.
Britain's interior ministry declined to comment on leaked information. A spokeswoman said the data would be published in the coming months.
Britain and China have been arguing for months about what London and Washington say is an attempt to silence dissent in Hong Kong after pro-democracy protests in 2019 and 2020.
The British flag was lowered over Hong Kong when the colony was handed back to China in 1997 after more than 150 years of British rule - imposed after Britain defeated China in the First Opium War.
Hong Kong's autonomy was guaranteed under the "one country, two systems" agreement enshrined in the 1984 Sino-British Joint Declaration signed by then Chinese Premier Zhao Ziyang and British Prime Minister Margaret Thatcher.
China says Britain's views on Hong Kong are clouded by an imperial hangover and that the territory needs the national security law to counter damaging unrest.
China and Hong Kong have said they will no longer recognize the BNO passport as a valid travel document from Jan. 31. BNO status was created by Britain in 1987 specifically for Hong Kong residents.
The British government has forecast the new visa could attract more than 300,000 people and their dependants to Britain. Beijing said it would make them second-class citizens.