Britain is one of several European Union countries insisting that Swiss banks share information about EU citizens' accounts.
Brussels is keen to clamp down on tax cheats who use non-EU countries to squirrel away savings.
But the Swiss government says banking secrecy is "not negotiable" and has instead proposed a withholding tax on EU citizens' interest earnings in Switzerland. The Swiss would then pass the tax revenue back to the EU citizen's home countries.
Couchepin's visit to London was part of a concerted campaign to win support for such a tax and comes ahead of a crucial meeting of EU finance ministers in Brussels on Tuesday.
For all sides, the clock on the taxation issue is starting to run down.
Faced with a need to halt on-going tax evasion, the EU has set a deadline for the end of this year to agree on a pan-European taxation system.
A basic aim is the automatic sharing of information about citizen's banking activities.
However, for that to work, Brussels needs the cooperation of so-called third states.
Switzerland has long been seen as the lynchpin. If Bern agrees to share banking information with EU tax commissioners - as part of a series of bilateral agreements currently under negotiation - other non-EU countries and tax havens would be expected to follow suit.
So far, however, the Swiss have stood their ground - to the infuriation of Europe's biggest powers.
Last month Britain, along with Germany and France, called for the drawing up of possible sanctions against Swiss financial institutions.
While serious, the sanctions threat is unlikely to be enforced, as several EU members have said they won't support the plan.
They include Luxembourg, which also has banking secrecy and which is one of Bern's staunchest allies in the tax dispute.
Banking secrecy can fit "moral obligations"
Luc Frieden, the Luxembourg treasury minister, recently visited Lucerne for a business conference, where he explained his country's stance.
"We do not say that we want to uphold [Swiss] banking secrecy," Frieden told swissinfo.
"We want to find a way to tax - in a fair and reasonable way - those who want to put their savings in another country than their country of residence," he said.
"That is why we are [supporting] a withholding tax - so that the Luxembourg citizen can put his money in Germany, where he will be taxed there and we get some money back.
"If you look at banking secrecy in that way, it complies with all moral obligations."
UK must compromise
Frieden said the EU was on the verge of finding a solution, although resistance by Britain was slowing progress.
"The breakthrough is on the table," he said.
"It is only for certain countries, especially the United Kingdom, to accept the coexistence model [of information sharing and a withholding tax].
"We have been discussing the taxation of savings for more than ten years, and I think it's a failure for Europe if we can't agree... but that requires some countries to compromise."
Switzerland's refusal to back down on banking secrecy has seen it accused of being a friend of the tax cheat, and in one case, of behaving like the "Iraq of the Alps".
But despite the vitriol, there are indications it may get its way.
EU and Swiss negotiators held talks last week to discuss details of the withholding tax proposal... a clear sign that more and more EU members are coming around to the idea.
swissinfo, Jacob Greber