A full 74 percent of people surveyed said they were opposed to ripping up the agreements, which cover everything from free movement of people to air traffic to trade barriers, according to the poll in the SonntagsBlick newspaper.
Only 19 percent of the 1,002 people questioned last week by Isopublic pollsters wanted Switzerland to completely turn its back on EU relations, while seven percent were undecided, the poll showed.
The survey was published a week after Swiss voters narrowly approved curbing immigration from the EU, with 50.3 percent of voters deciding to void a pact giving equal footing to EU citizens in the Swiss labour market.
The vote in favour of the "Stop Mass Immigration" initiative put forward by the rightwing populist Swiss People's Party (SVP), could set in motion a so-called "guillotine clause", freezing the entire package of Swiss-EU deals, including on trade.
Switzerland is not a member of the EU, but counts the bloc as its main trading partner and its economy is closely intertwined with European and global markets.
The EU voiced outrage at the referendum result, insisting that Switzerland cannot pick and choose which accords to implement and pledging to review "relations as a whole".
The February 9th vote targeted a 1999 agreement under which Switzerland adopted EU freedom of movement rules, which was broadened in 2008 when the country also joined the bloc's Schengen Accord that allows for passport-free travel.
But if Bern and Brussels fail to agree on a way forward and Switzerland voids the free movement accord, the "guillotine" clause comes into force, impacting agreements coving trade and economic ties, market access, and agricultural produce to name a few.
On Saturday, the Swiss justice ministry said it declined to sign a deal opening market access to people from Croatia, the EU's newest member state.
The referendum measure on immigration binds Bern to renegotiate the entire EU labour market deal within three years.
The current rules stay in force in the meantime.
The Swiss government aims to present a plan on how to implement the Swiss people's will on the issue before July, and to present a proposed law by the end of the year.
SVP chief Toni Brunner said in an interview with Schweiz am Sonntag published Sunday that Switzerland needed to move faster to put measures in place.
This was because "a new wave of massive immigration" was about to hit as the EU extends its freedom of movement to all its eastern members, he said.
The head of the Swiss Socialist party, Christian Levrat, told the SonntagsZeitung weekly he wanted a new popular vote in the country on the future of the bilateral deals with the EU.
He said that voters had assumed the proposal would not jeopardize existing agreements.
"If that is not the case, we have a new situation that justifies a vote," he said.