Swiss offer EU solution for Britain
The former Swiss president has said the UK should work with her country to find a way to balance freedom of movement with single market access.
Micheline Calmy-Rey said the two countries shared a common purpose out - or planning to be out - of the European Union.
But both still want a good trade relationship with the union.
She also urged the European Union to become more "flexible" on the issue of immigration or risk "exploding".
Its "very rigid" approach to free movement was not the correct response to the new pressures it faced following the UK vote to leave, she said, adding that Switzerland had found it very difficult to negotiate with the EU on the issue.
In 2014, Switzerland voted against free movement in a referendum.
"Switzerland and Britain have in common the same question - that means the question of freedom of movement and [the need] to find a solution in order to respond to our population to limit migration," Ms Calmy-Rey told the BBC.
"The question is what political price Great Britain and Switzerland are willing to pay to limit migration.
"Do they have to pay with access to the big market - totally, partially or not at all?
"And it all depends on the negotiations you will have and we will have with the European Union.
"I think not only Great Britain and Switzerland have this problem, I think other member states inside the European Union have this problem of freedom of movement."
The BBC understands that Swiss trade diplomats have visited Britain and met Whitehall officials to open lines of communication.
Although Theresa May and the Chancellor, Phillip Hammond, have spoken about wanting a "bespoke deal" with the EU for Britain, the similarities between the UK position as it plans to leave the EU and Switzerland are striking.
And means there will be conversations between the two countries.
Switzerland rejected membership of the European Economic Area which allows for single market access in 1992 and is the only non-EU country in Europe which has its own bilateral trade deals with the union.
As part of those trade deals - which number more that 130 - it has agreed to open borders with the EU, an issue which has become increasingly controversial as people complained that EU citizens working in Switzerland were earning money in valuable Swiss Francs and then exchanging them for euros.
There were also concerns that the Swiss population was increasing too quickly for the country's infrastructure.
In the 2014 referendum, the Swiss voted to impose quotas.
Since then, Ms Calmy-Rey said the European Commission had refused to move on the issue of limiting freedom of movement while retaining tariff-free access to the single market.
But with Britain now making similar arguments to Switzerland, pressure is growing on the EU to compromise.
Ms Calmy-Rey, who was president of Switzerland between 2007 and 2011 and foreign secretary between 2003 and 2011, has considerable experience of EU trade negotiations.
She said the EU had to have a more "diverse" approach to non-EU countries as it would not want to see a large European economy like the UK "go towards" Asia or America.
"I think we could collaborate together - it would be fruitful," she said, arguing that many countries in the EU had similar concerns about freedom of movement, at present one of the four founding principles of the union.
"You have one vision [in the EU] which wants more integration, more solidarity with migrants, more freedom of movement and more integration of economic policies, of fiscal policies," Ms Calmy-Rey said.
"And you have other countries that do not want all of that.
"If the European Union doesn't want to take the risk of exploding we have to take into account these two visions and perhaps find a third way."
With little movement from the EU, the Swiss parliament has now suggested a unilateral approach.
In proposals announced last week, a parliamentary commission said that a "safe-guard" clause could be used to give people living in Switzerland priority in the jobs market over people coming to the country from the EU.
That is not likely to be sufficient for the UK, which, Swiss diplomats admit, will have far more leverage in negotiations with the EU because of size of the UK economy and the importance of London as a banking and insurance centre providing access to finance for EU businesses.