Mario Gattiker made the comment to the newspaper Basler Zeitung after the paper obtained a leaked document from the Federal Migration Office (SEM) suggesting as many as 10,000 people could arrive in Switzerland in November alone.
Gattiker wouldn't confirm the prediction, saying instead: “We are registering the same ongoing high level of application numbers as before".
However right-wing MP Ueli Maurer leapt on the prediction. Speaking on the SRF programme 10 vor 10, he said: “If you look at the whole situation in Europe then it would certainly be conceivable that Switzerland will receive 10,000 this month.”
Confirmed figures from the SEM show that 4,544 new arrivals applied for asylum in Switzerland in September, 645 more than the previous month.
From January to September the country received a total of 24,212 asylum seekers, a 34 percent rise on the same period last year.
Basler Zeitung estimates the figure for October – so far unconfirmed by SEM – to be 4,700.
According to the paper, Switzerland is becoming more popular with Afghan migrants, with 1,533 seeking asylum in the alpine nation in October alone, a significant rise when compared with the total 1,929 Afghan arrivals in the nine-month period from January to September.
Gattiker told the paper the influx reflects Germany's policy of favouring Syrian refugees over Afghans.
Speaking to the NXP news agency, the president of the country's centre-right Liberals party Philipp Müller said the country must react.
“The confederation must very clearly say that Afghan refugees are only provisionally welcomed in Switzerland.”
According to NXP the government has budgeted for 24,000 refugees in 2016, at a cost of 1,47 billion francs.