The Bank recently received plans from Asian British Business Community, a Hinduja-backed group, to establish the British Asian Bank.
The Hindujas, with an estimated wealth of more than pounds 1bn, rank in the top 10 of Britain's wealthy. From a base in London's Haymarket, Srichand and Gopichand Hinduja - two of four brothers - run a secretive empire spanning manufacturing in India, oil trading in the US and Middle East, airports and power stations.
In 1990, Gopichand was one suspect named in a criminal report by police at India's Central Bureau of Investigation in a continuing inquiry into pounds 30m of kickbacks allegedly paid by Swedish arms firm Bofors to secure a pounds 775m howitzer deal in 1986.
Under the Banking Act, the Bank has to satisfy itself that backers are "fit and proper" to hold a UK banking licence.
The application comes at a sensitive time for the Bank as it is being sued for alleged regulatory lapses over the pounds 10bn collapse of the Bank of Credit and Commerce International. BCCI marketed itself widely to the Asian business community and left thousands in the lurch when it was closed in 1991.
The Hindujas already own two banks, in Bombay and Switzerland, but the Bank says that would not neccessarily ease their path in the UK.
"It would make a difference whether individuals had been 'implicated' or actually charged [in a criminal case], but obviously probity of individuals is important," a Bank source said.
The Bofors scandal rocked India and contributed to the 1989 election defeat of the late prime minister, Rajiv Gandhi.
Bofors has always said the payments were legitimate commissions, but has supplied details of six Swiss bank accounts which have since been frozen.
The Hindujas, meanwhile, have denied any involvement whatsoever. Both Gopichand and Srichand, together with Geneva-based brother Prakash and a Hinduja firm, Jubilee Finance, have since appealed in Swiss courts to stop bank account details being released.
They have already lost one appeal, but a further one over specific documents is still continuing, though Indian investigators are hopeful of an early result. "We hope that the Swiss cantonal and federal courts will not take that long now over matters of detail. They should decide in three to four months," one Indian police source said.
The brothers are currently travelling in India and could not be contacted, according to the office of their London company, Sangam Ltd. Last year, however, they said they had been forced to appeal in Switzerland to ensure that other Bofors payments did not get "mixed up" with the gun contract inquiry.Reuse content