The survey by the gfs.bern polling institute showed that 52 percent of voters opposed the plan, while 42 percent were in favour.
With just six percent of voters still to make up their minds, a turnaround by supporters of the fighter purchase appeared unlikely, the polling institute said.
Approved by the government in 2011 and backed by parliament last September, the fighter deal cannot be blocked as such in the referendum.
But opponents have been able to contest legislation that allowed the purchase to be funded by tapping an annual 300 million Swiss francs ($342 million) from the neutral country's military budget over 10 years.
The Swiss defence ministry has said that in the event the people reject the financing legislation, the decision will be respected, and that there is no back-up funding plan.
Switzerland opted for Saab's Gripen jets over the Rafale made by French company Dassault and the Eurofighter of pan-European firm EADS.
Supporters underline that in exchange for the sale, Saab and its engine supplier are contractually bound to sign business deals with Swiss firms worth 2.5 billion francs over the next decade.
Opponents dismiss that, and say that the 3.1-billion-franc price tag for the jets could end up tripling because of operating and maintenance costs as well as future upgrades.
The anti-deal coalition is steered by the left-leaning Socialists and Greens, plus anti-militarists who last year lost in a referendum in which voters bucked a European trend and kept their conscript army.
The Gripen's adversaries also include economic liberals opposed to the price.
Referendums form the cornerstone of Switzerland's system of direct democracy, and the campaigners forced a plebiscite by mustering more than 65,000 signatures from voters.
The Gripens are meant to replace Switzerland's three-decade-old fleet of 54 F-5 Tigers, built by US group Northrop.
The Swiss air force also has 32 F/A 18 Super Hornets built by US company McDonnell Douglas, purchased in 1996.
The military notes that with a combined fleet of 54 Gripens and Super Hornets, Switzerland's fleet would lag far behind the 300 jets in service in the early 1990s.