This week's vote on tightening the asylum laws is set to be close, according to the last opinion poll before the ballot.
Some 43 per cent said they would vote "yes" to the rightwing proposal with 37 per cent against, but the gap has narrowed considerably in recent weeks.
The Swiss are to vote on Sunday on an initiative by the Swiss People's Party, which would limit asylum applications and cut benefits to those whose requests have been turned down.
The previous GfS research institute poll, in October, showed support for the initiative at 57 per cent with only 24 per cent saying they would vote "no".
But in the last poll before the ballot on November 24, support had fallen to just 43 per cent, while the number saying they would vote against was up to 37 per cent.
But crucial to the vote are the 20 per cent who are still undecided about how they will vote, one per cent higher than in October.
The survey's compilers, who polled 1,297 people, said the results showed the government's campaign against the proposal had made significant progress.
Last month's poll was conducted before the government had laid out its opposition to the initiative and the yes campaign was in full swing.
Tighter asylum laws
The proposal would prohibit asylum seekers from filing an application in Switzerland if they had passed through a so-called "safe" country first.
Since Switzerland it surrounded by "safe" countries, the effect would be to limit substantially the number of applications. The federal authorities say around 95 per cent of asylum seekers arrive in Switzerland through Austria, France and Germany.
In October's poll, support for the initiative was strong among supporters of three of the four main parties. Seventy-two per cent of People's Party members and 69 and 55 per cent from the centre-right Radicals and Christian Democrats said they would vote for the proposal.
This month's results show that although People's Party supporters have rallied behind their initiative with 78 per cent in favour, only 40 per cent of the centre-right Radicals and 28 per cent of Christian Democrats would now join them.
The centre-left Social Democrats have increased their majority against the initiative from 56 to 64 per cent.
The government has been trying to persuade voters to reject the asylum initiative saying it would be difficult to enforce and would make it harder to distinguish bogus asylum claims from genuine ones.
The justice minister, Ruth Metzler, said at the launch of the no campaign that the People's Party's proposal would be counterproductive, expensive and against Switzerland's humanitarian tradition.
"We want to be tough when it is necessary to send people back, but we also want to be humanitarian and protect people when they need it, " she told swissinfo.
The proposal has also come under harsh criticism from the United Nations High Commissioner for Refugees, Ruud Lubbers. He condemned the People's Party's campaign as "misleading", saying it could lead to some of the toughest asylum restrictions seen in any industrialised nations.
"If the Swiss people vote 'yes' to the initiative it will effectively mean that any refugee arriving in Switzerland by land will rejected straightaway, however well founded their request for asylum," said Lubbers.
His comments were dismissed by the People's Party as "interference" in the country's internal affairs.
Human rights groups have also pointed to an increase in racism and xenophobia directed towards blacks and asylum seekers ahead of the vote.
No firm grip
Despite the gains made by the no campaign, just over 70 per cent of those polled thought that the government did not have a firm grip on the asylum situation and almost three quarters of them found the costs of asylum too high.
But the survey found that the strongest argument against the initiative was concern that a tougher approach would conflict with Switzerland's humanitarian tradition.
The number of yes votes was highest among men, older people and non-graduates, while the no votes were found predominately among women, younger people and those with a degree.
swissinfo, Isobel Johnson
In November's poll 43% would vote for tightening asylum laws, 14% down from October.
37% would vote against, up 13% from October.
20% are still undecided, up 1% from October.
Sunday's vote is expected to be close with the undecided voters playing a crucial role in the outcome.