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.
Search for missing man in Rhône
Emergency services are searching the Rhône in Geneva tonight for a young man who failed to resurface after jumping into the river with three friends.
Geneva worst for traffic
Geneva is now the most congested city in the country, with Lausanne in third place.
Geneva's cross-border compensation payment to France reaches record CHF 411m
Geneva will transfer a record CHF 411 million to France this year under the Financial Compensation scheme, up from CHF 396 million last year.
Crans-Montana bar accountant facing criminal allegations
The investigation into the deadly Crans-Montana bar fire has taken a sharp new turn.
Geneva tops list for cocaine use
Cocaine use in Switzerland continues to rise, according to a new study by Addiction Switzerland.
Bol d'Or rocked by controversy
Controversy has overshadowed the 87th Bol d'Or sailing competition this weekend.
