More than 100 people from ethnic minorities have told the BBC their bank accounts have been closed without explanation.
Banks say they are under an obligation to close accounts if they suspect money laundering.
Rushanara Ali, MP for Bethnal Green & Bow, says any broad patterns showing discrimination should be investigated.
But the British Bankers Association says its members do not discriminate on the basis of ethnicity.
Sameer recently got married in Pakistan and relatives in the UK paid several thousand pounds into his account as a wedding present.
He believes those payments are the reason his TSB account has been closed, as he told Radio 4's Money Box programme: "It was £5,000 to £6,000. After three days I tried to take some cash out and the machine swallowed my card.
"I received a letter after two days saying they were going to close my account in 60 days. I explained everything but they said that wasn't good enough. I didn't hear anything about white people's accounts being closed. So far I've heard about people whose background is Asian."
TSB said it did not discriminate against customers in this way: "We treat all our customers equally and never take the decision to close someone's account lightly. On the rare occasions where we do close a customer's account, it is only ever done when they are in breach of our terms and conditions."
The reasons why Sameer's account was closed are still unclear.
One hundred and two customers with names which were not typically British whose accounts were closed have contacted Money Box since the start of last year out of a total of 150 emails. This is not a scientific sample but it included customers of all the big high street banks.
Chukwudi also contacted Money Box after his bank account, which he had had for six years, was closed by Royal Bank of Scotland.
His parents are from Zimbabwe but he was born in Germany and has lived in the UK for 10 years.
He says he has no idea why he has had his banking facility taken away: "I asked them what was happening as I'd been banking with them for quite a long time. They said no, they're not going to let me know."
The British Bankers Association said its members did not take account of ethnicity when deciding whom to offer services to: "Banks operate in line with UK and international regulations for financial crime and financial services; they do not discriminate on the basis of ethnicity.
"There are certain communities that are being targeted by criminals to help facilitate money laundering. Banks are making considerable efforts to minimise any adverse actions on genuine customers and the BBA is seeking to engage the relevant authorities on the issues."
Barry Vitou is head of corporate crime and investigations at the law firm Pinsent Masons. He advises banks and businesses on complying with money-laundering legislation. He says banks have little leeway in how they can act, and what they can say to customers.
He said: "They have been told time and time again in recent years that money-laundering procedures, systems and controls that they have aren't up to scratch. They're in between a rock and a hard place. If they are suspicious, they have an obligation to report that suspicion under the UK money-laundering legislation, and they can't tell the customer. "
Ms Ali said that if evidence emerged of widespread discrimination, there should be an investigation: "What we need from any bank that decides to cancel people's bank accounts unilaterally is an explanation and some transparency.
"If there is a widespread pattern of refusal of banking services, the Equality and Human Rights Commission should be looking at this issue."
Dan Plant sits on the Financial Services Consumer Panel, which has been investigating the issue. He said: "The bank acting as effectively judge, jury and executioner here is a subversion of natural justice."