If Switzerland does not back the draft pact, however, the European Commission will not extend the recognition of the SIX Swiss Exchange - the country's main stock exchange - and other trading venues beyond this year, EU sources said following a meeting of EU diplomats on the issue on Wednesday.
The Swiss cabinet is expected to decide on Friday on whether to back the draft agreement reached with the Commission on future bilateral ties.
"It's deal time. Playing for time wouldn't change anything. The draft will not be changed anymore," an EU official said, staking out a hard line that is likely to be replicated in talks with Britain on its future relations with the bloc after it leaves the EU in March.
Under the draft deal, which would govern EU-Swiss relations currently regulated by about 120 accords by sector, Switzerland would automatically adapt its migration and social security rules to changes in EU legislation, EU diplomats said.
The EU wants the overall framework agreement before it extends recognition of Swiss exchanges beyond the end of this year, when it will otherwise expire.
Financial operators from countries that are not members of the EU can operate in the bloc under a so-called "equivalence" regime which recognises that foreign-country market rules are compatible with EU legislation.
This regime, which would also concern British firms after Brexit, is granted for limited periods and could be suspended by the EU anytime if significant changes are made in foreign rules.
The SIX bourse has repeatedly warned that it risked losing much of its exchange business if no EU-Swiss deal was reached by the end of the year on future relations.
Switzerland has been for decades integrated in the EU market but is not a member of the 28-country bloc.
But the chances of a deal remain slim.
Political sources in Switzerland said the Swiss cabinet does not have a majority to press ahead with a treaty deal.
A Swiss government spokesman declined to comment, adding it "will inform the public after having taken its decisions".
EU officials also recognised that Friday's cabinet meeting might not be definitive, but they insisted that there will not be a new offer from Brussels.
EU states supported the Commission's stance, an EU official said.
A Commission spokesman declined to comment.
Under the draft deal currently on the Swiss government's table, an arbitration tribunal would resolve bilateral disputes but the European Court of Justice would have the final say on matters concerning the application of EU laws, diplomats said.
The Swiss would also contribute to the EU budget and provide financial support to poorer regions in the EU in a way that is proportional to the benefits the Alpine country draws from having unhindered access to the EU market.
(Reporting by Francesco Guarascio in Brussels; Additional reporting by Michael Shields in Zurich; Editing by Alissa de Carbonnel)
By Francesco Guarascio