The EU wanted Switzerland to approve a broad deal that would simplify relations, currently defined by a messy set of some 120 bilateral accords, and said it expected an answer Friday to the final terms on the table.
"Significant progress has been made" on reaching a deal, Swiss President Alain Berset told reporters.
But he added that "there are still differences on very important issues" and the Alpine nation's government needed time for broader consultations.
Swiss Foreign Minister Ignazio Cassis said those consultations may take place in the spring.
The draft agreement on the table covers an array of issues, including the thorny question of free movement of people.
Seeking leverage over Bern, the EU has threatened to not renew the so-called 'equivalence' status of the Swiss stock exchange unless the broader deal was agreed.
Equivalence allows EU-based trading platforms to buy and sell Swiss stocks. If Brussels takes that away, the Swiss exchange faces a huge hit from trade volume losses.
The European Commission said it "respects the will of the Federal Council to consult all the parties concerned" but called for "speedy" negotiations.
For Switzerland – and its steadily robust economy – the consequences of dismissing the EU deadline are not yet clear.
Switzerland last week activated a plan to protect its stock exchange should the EU revoke equivalence guidelines.
EU-based platforms that want to trade Swiss stocks like Nestle are now being forced to apply for registration with Switzerland's authorities.
The idea is to block platforms from trading Swiss stocks within the EU and thereby force traffic back to the Swiss exchange.
The Bern-Brussels relationship suffered a heavy blow in 2014 when Swiss voters backed a proposal calling for the re-introduction of migrant quotas, which could have limited the number of EU citizens working in Switzerland.
The Swiss parliament in 2016 approved a modified version of that plan in an attempt to mend fences.
Polling shows that Swiss voters remain deeply divided about how closely to align with the EU.
Whatever the government ultimately decides will almost certainly be put to a referendum, as part of Switzerland's direct democracy system, creating further uncertainty on the terms of a final deal.