The Basel rules are the world's main regulatory response to the 2007-2009 financial crisis, aimed at preventing a repeat crisis where banks had to be bailed out by their governments.
But both the United States and the EU missed the January deadline for the start of a six-year phase-in of the new regime - the EU was not ready, and the United States in December announced a delay for further consultation.
Barnier said on Thursday he and Federal Reserve Governor Daniel Tarullo "agreed it was essential that effective implementation of the new regulatory framework should be done as soon as possible".
The EU is expected to formally implement Basel next January, a year late but despite the delay the biggest EU and US banks already meet or exceed the new capital requirements that don't come fully into force until 2019.
Separately, Tarullo told US lawmakers he hoped the rules would be completed in the spring.
An EU official said officials meeting on Thursday also backed proposals for a capping of bankers' bonuses to no more than the equivalent of their fixed salary, a so-called 1:1 ratio, although the bonus could be doubled if backed by a simple majority of shareholders at a meeting where two thirds of shares are represented.
The move was proposed by Ireland in its role as the current president of the EU in seeking to agree a negotiating mandate for talks with the European Parliament next Tuesday on a final deal on the bonus rules.
"There was broad consensus in favour of the Irish presidency's approach, with the notable exception of Britain," the official said.
"We received support from member states for our overall approach," a spokeswoman for the Irish presidency said.
Britain suggested the 1:1 ratio could be between fixed pay and cash, with the possibility for shareholders adjusting this by a majority of votes cast.
Some states, including Germany, said Britain's suggestion could feed into next week's talks with lawmakers.
The shift by EU states towards the idea of capping bonuses brings them closer to the European Parliament's desire to introduce what would be the world's toughest curbs on bankers' pay, though other tricky issues remain to be resolved.
"I am not sure we will get a deal on Tuesday," said Sharon Bowles, who chairs the Parliament's economic affairs committee and will take part in next week's negotiations.