Support is slipping for the Swiss People's Party initiative to cap the population at 10 million, ahead of the 14 June federal vote.
Two major polls, from SSR and Tamedia, now both put opposition at 52%, with the yes camp at around 46%.
Pollsters say undecided voters are moving towards rejection as the campaign highlights the economic risks of restricting immigration.
Voters head to the polls in just over a week.
Swiss Post announce job cuts
Swiss Post will eliminate up to 110 positions in 2027, blaming shrinking letter volumes and fewer over-the-counter transactions
Autumn arrives in Switzerland
The heatwave that has gripped Switzerland since May finally breaks tomorrow
Driver warning in Vaud
Drivers in western Vaud face four nights of disruption on the A1 next week.
Mixed results in school classrooms
Swiss 15-year-olds still rank among the world's best in maths and science, but their performance in reading and maths is sliding
Good news for household bills
Swiss households will pay a little less for electricity next year
Fuel reserves now in use
Switzerland is dipping into its emergency fuel reserves for the first time this year to keep petrol and diesel flowing
