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.
Washington proposes 12.5% tariff on Swiss exports
Washington has proposed an extra 12.5% tariff on Swiss exports as part of a sweeping new offensive against 60 economies.
National Council backs sweeping powers for intelligence service
The National Council has voted to widely expand the powers of Switzerland's Federal Intelligence Service, citing growing terrorist, cyber and foreign-interference threats.
Crans-Montana mayor refuses to resign over New Year disaster
Crans-Montana president Nicolas Féraud has broken 126 days of silence to confirm he will not resign over the New Year's Day disaster at the Constellation building.
Parenting tops Swiss mental health stressors
Parenthood is the leading factor weighing on the mental health of adults in Switzerland - more than work or money - according to a new AXA study.
Vaud promises 'resource spaces' in every school by 2028
Vaud has unveiled a new plan to help teachers cope with disruptive and high-needs pupils, after rising complaints that classroom misconduct is undermining lessons.
Montreux school buys CHF 40 million villa for boarders
The St George's International School in Montreux has bought a historic Belle Époque villa for CHF 40 million, planning to convert it into a prestigious boarding house.
