Modi, whose celebrity clients have included actress Naomi Watts and Kate Winslet, is accused of involvement in a $1.8 billion scam against Punjab National Bank (PNB), India's second-largest state-run lender.
The 47-year-old has been on the run since the scandal broke in February, rocking India's corporate world.
The Financial Times newspaper cited officials in India and Britain as saying that Modi had sought asylum in Britain for what he called "political persecution".
Indian officials were not immediately available to comment.
A spokesman for Britain's Home Office said: "We do not comment on individual cases."
It comes as India's government seeks the extradition of liquor tycoon Vijay Mallya over unpaid loans to his beleaguered Kingfisher Airlines after he absconded to Britain in 2016.
Indian investigators are probing allegations that Modi and his uncle and business partner Mehul Choksi, also a diamond merchant, defrauded PNB of 2.8 billion rupees ($43.8 million).
This figure is said to be just a part of the total losses.
PNB said in February that Modi and Choksi had defrauded it by raising credit with international branches of other Indian banks, using illegal guarantees provided by rogue PNB employees.
Investigators have arrested 25 people for enabling the defrauding of the bank with forged documents.
Modi, the third generation of his family to go into the diamond trade, is worth $1.73 billion according to Forbes, placing him 85th on India's rich list.
His high-end eponymous Nirav Modi brand has stores in several of the world's major cities and boasted of celebrity clients including Watts, Winslet, and Bollywood star Priyanka Chopra.
India's government is trying to reduce the crippling debts of the country's embattled state banks, including PNB.
It recently announced a $32 billion recapitalisation plan to help them clean up their books ahead of the general election in 2019.
Last month PNB posted the largest ever quarterly loss for an Indian lender.
It said it had incurred a record 134.17 billion-rupee net loss for the January-March quarter, having posted a 2.62 billion-rupee profit in the same period last year.
The losses were largely because it had to set aside funds to pay the other banks for the illegal guarantees issued in the scam.
Eyewitness? Submit your stories now via social or:
Email: eyewitness@pulse.ng
![The woman was recorded, among others, while bathing [BI]](data:image/svg+xml;charset=utf8,%3Csvg%20xmlns%3D'http%3A%2F%2Fwww.w3.org%2F2000%2Fsvg'%20width%3D'300'%20height%3D'172'%20data-ring-placeholder%3D'1'%3E%3C%2Fsvg%3E "The woman was recorded, among others, while bathing [BI]")
