The Competition and Markets Authority (CMA) said customers had not seen enough benefit from efforts to open up the market.
It said it was minded to launch a full-scale market investigation but has given the "big four" banks - Barclays, HSBC, Lloyds Banking Group and Royal Bank of Scotland - the chance to come up with their own solutions before a formal decision in the autumn.
The CMA said it had found that "essential parts of the UK retail banking sector lack effective competition and do not meet the needs of personal consumers or small and medium-sized enterprises (SMEs)".
It follows two studies in collaboration with City watchdog the Financial Conduct Authority (FCA) into the £8 billion personal current account market and the £2 billion SME current account and lending sector.
The CMA found that concerns remained about competition not effectively serving customers despite measures to make authorising new banks simpler and faster, to make account switching easier and to improve transparency.
It found that it was still too hard for newer and smaller banks to enter the market or expand, with much business remained concentrated in the hands of a few.
There was "very little movement" in the market share of the largest banks - other than as a result of mergers and acquisitions - and many customers saw little difference between the largest banks in the services they offered, the CMA said.
It added that levels of shopping around and switching between banks remained low and that very limited gains had been made by those with the highest levels of customer satisfaction.
This was "not what would normally be expected in well-functioning, competitive markets", the CMA said.
The watchdog also said it was hard for customers to make comparisons between lenders, particularly on complex overdraft charges, limiting banks' incentives to compete, and possibly resulting in higher overdraft charges.
Meanwhile, the extent to which the firms used "cross-subsidy" between different retail banking products and different customer groups using "free-if-in-credit" accounts may also "distort competition".
CMA chief executive Alex Chisholm said: "Competitive personal and SME banking markets are essential to households and businesses throughout the country, and to the success of the UK economy.
"However, our studies have found that, despite some positive developments, significant competition concerns remain which mean that customers may not be getting consistently good service and value from their banks.
"Our provisional view is that a full market investigation by an independent expert CMA group is necessary to look at this market in detail and identify appropriate measures if competition concerns are found.
"However we very much welcome views, which we will carefully consider, before taking a final decision."
Mr Chisholm told the BBC Radio 4 Today programme: "Small businesses say they are not happy with the choices they face and the service they are getting, and in the personal market as well the Big Four banks have satisfaction ratings below 60%.
"Banks are adapting to the changes in the market place in the aftermath of the financial crisis, there's no question about that.
"We say they need to adapt so as to be better focused on their customers and there's no excuse for not doing that.
"We're not suggesting collusion. What we are suggesting is that they are not putting each other under enough pressure, there's not enough rivalry there to be really falling over themselves to serve the customers and come up with new innovations and the best value and the best service they can provide.
Business Secretary Vince Cable said: "My long-standing concerns about the state of competition in UK banking are well documented, so I welcome the CMA's announcement today.
"This is an issue that really matters for the real economy - constraints on banking competition mean less choice for both consumers and small businesses seeking finance to grow.
"This Government has already acted to improve competition, by setting up tough new regulators, establishing the British Business Bank to make the business finance market work better, and making it easier for new players to enter the market.
"But the CMA's initial decision to conduct a full market investigation is a very significant further development, and I very much look forward to their final decision in the autumn."
Shadow chancellor Ed Balls said: "Ed Miliband and I have repeatedly called for an inquiry into bank competition, so it's welcome that the Competition and Markets Authority is now set to start this work later this year.
"Ministers claim there is no problem to solve, but everyone else recognises that we have a lack of competition in our banking sector.
"As we said earlier this year, in the next parliament we need to see at least two new challenger banks and a market share test to ensure the market stays competitive for the long term."
Richard Lloyd, executive director of consumer group Which?, said: "For too long customers have been getting a raw deal from the biggest high street banks, so a full inquiry into the current account market is welcome if long overdue.
"While there have been encouraging signs of change from some banks, we need to see a revolution in customer service and much better, easily comparable products if more people are to be convinced that it's worth switching accounts.
"The CMA must now get to the bottom of why this market is not working for consumers, but the banks should not waste any time in making changes to put the interests of their customers first."
Christopher Woolard, director of policy, risk and research at the FCA, said: "Small businesses are the lifeblood of the British economy, employing more than half of those in work in the private sector, which is why it is vital that they have access to a banking market that works for them."
John Longworth, director-general of the British Chambers of Commerce, said: "Businesses want more competition, choice and transparency in the banking sector, so the provisional decision by regulators to conduct a full-scale investigation into business banking is the right one."