Swiss ministers have had "very positive" talks in Washington, US officials say, in a bid to cut US President Donald Trump's steep 39% tariffs on Swiss exports to the US – the highest in Europe.

One senior US official said the Swiss were "very aware" of their trade deficit and were prepared to address it - hopefully leading to a "meaningful reduction" in tariffs.

Swiss economics minister Guy Parmelin, meanwhile, said the two sides had "clarified almost everything" in the talks.

Switzerland been trying to bring down the tariff rate - which has hit the country hard - for months.

Swiss industry chiefs came to the Oval Office on 4 November bearing gifts, including a Rolex gold watch and a specially engraved gold bar from Swiss-based gold refining company MKS.

Initial attempts by Swiss President Karin Keller Sutter to change Trump's mind fell on deaf ears. However, a private business initiative last week may have made all the difference. Trump's response to the Swiss president's bid was that she "was a nice woman, but she did not want to listen".

But after last week's private business initiative Trump said a deal was being worked on to bring the tariffs "a little bit lower… I haven't set any number".

For Swiss industry, that deal could not come soon enough. Tech exports to the US are down 14.2% on the third quarter of last year, according to latest statistics - a dramatic fall since the tariff hike was imposed in August.

Some figures in Swiss business, particularly those trading in luxury goods, gold, or commodities, already had contacts in Trump's circle.

In September, Trump appeared at the US Open tennis final in the Rolex VIP box hosted by the Swiss watch company's chief executive Jean Frédéric Dufour.

The president, apparently guessing what was going on, even asked if Dufour would have been there if Trump had not slapped such steep tariffs on Switzerland.

Last week Dufour met Trump again, this time in the Oval Office, along with fellow business leaders including Johann Rupert from luxury goods maker Richemont and Marwan Shakarchi from MKS.

It is quite normal nowadays for any leader heading to the Oval Office to come bearing a gift.

UK Prime Minister Sir Keir Starmer brought an invitation from King Charles for a lavish state visit. German Chancellor Friedrich Merz offered a framed copy of Trump's German grandfather's birth certificate.

Requests for confirmation of the gifts to the two Swiss companies involved brought a "no comment" from Rolex and MKS.

But days after the meeting, Trump was pictured in the Oval Office with what looked very much like a Rolex "Datejust" desk clock, produced by the company as a collector's item, and worth tens of thousands of dollars.

A White House official confirmed the two items had been given to Trump.

The US president receives thousands of gifts every year and they then become US property, deposited with the National Archives and filed annually by the state department.

They are eventually transferred to a presidential library. Some gifts can be kept but presidents have to pay federal taxes if they do not come from a close relative.

In 1969, President Richard Nixon gently refused the gift of a Swiss Omega watch to commemorate the Moon landings.

Whatever happens to the Swiss gifts, Trump's stance towards the Swiss appears to be softening, telling reporters he is working on something "to help Switzerland".

Swiss economy minister Guy Parmelin and chief trade negotiator Helene Budliger Artieda, who travelled to Washington on Wednesday, are more hopeful than they have been in months, amid suggestions that 39% tariff may be reduced to 15% - the same as Switzerland's neighbours in the EU.

In return, promises from the Swiss pharmaceutical giants to build more production plants in the US are already on the table. It is also reported that Swiss International Airlines, whose fleet is primarily Airbus, may pivot towards Boeing.

But will it be enough? Swiss industry is waiting with bated breath. A number of Swiss companies have warned they will have to furlough staff if nothing changes.

The Swiss do have one more highly influential figure they can call on.

Fifa president and Swiss citizen Gianni Infantino, long a friend of Trump's, was reportedly urged by some Swiss parliamentarians to try to change the president's mind.

As part of preparations for next year's World Cup in the US, Canada and Mexico, Infantino visited the Oval Office in August bearing the trophy.

As the cameras rolled, he handed it to Trump saying he was "a winner". The president responded asking "can I keep it? That's a beautiful piece of gold".

Infantino has also announced a brand new Fifa world peace prize, to be announced in Washington DC on 5 December.

All bets are off as to who that might be.