Such a deal would come hot on the heels of a tax agreement recently reached between Switzerland and Germany that will net Berlin billions of Swiss francs and force the Swiss banking sector to clean up its act.
Odier, president of the Swiss Bankers Association, said talks with Britain were very advanced, though he added important points still have to be clarified.
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"I have no doubt that we will strike a deal with Britain very soon," Odier said. "I hope that the conclusion comes in the coming weeks," he said.
In an interview with Swiss newspaper Sonntag, Claude-Alain Margelisch, chief executive of the Swiss Bankers Association, said that the flat-rate withholding tax rates in the British deal would be in line with Britain's domestic tax burden.
Odier also said the German deal had had a "signaling effect" on France and Italy, and that he hoped new negotiations with these countries would be held.
"I am sure that other countries in Europe will also say that they would like to reach a deal with Switzerland," Odier said, adding the deal with Germany would likely cost Swiss banks about $642.9 million to implement.
Swiss banks have also come into the crosshairs of U.S. authorities.
In 2009, the Swiss government cut a deal with Washington to hand over the details of 4,450 UBS accounts in return for the dropping of a damaging lawsuit against the bank, while U.S. authorities are now also investigating Credit Suisse.
"We would welcome it if Switzerland and the United States were able to start talks quickly for a global solution," Odier said when asked about the problems facing Swiss banks in the United States.
Margelisch said it could not be ruled out that Switzerland would have to hand over details of accounts to the United States again.
"The American want client details. If this is delivered, then it clearly has to be done within the legal framework," Margelisch was quoted as saying in the interview.
(Reporting by Katie Reid; editing by Mike Nesbit)