The US ambassador initially said dual nationals were included in President Trump's 90-day immigration ban on citizens of Iran, Iraq, Syria, Libya, Sudan, Somalia and Yemen.
In a statement on Tuesday the Swiss foreign ministry said it had been talking to US authorities since Trump's executive order last Friday, seeking to “preserve the interests of Swiss dual nationals”.
As a result, the US embassy in Bern has now said that dual nationals can travel to the US if they have a valid visa in their Swiss passport, said the foreign ministry.
However dual nationals without a valid US visa cannot currently apply for one. The duration of this suspension of visa applications has not been communicated, it added.
Those affected can call a helpline run by the Swiss foreign office (058 465 33 33), which will pass on inquires to the US embassy, it said.
The clarification on dual nationals from the US authorities was also communicated to other European countries on Tuesday including Germany and Denmark.
In addition to the 90-day entry ban on seven Muslim-majority countries, Trump also suspended the US refugee resettlement programme for 120 days and banned all Syrian refugees indefinitely.
Trump's executive order on immigration has sparked global condemnation and protests.
On Monday a petition was launched in Switzerland calling for Trump to be banned from the country until further notice.