If the UK refuses then the Brussels club plans to hit back by blocking sectors from having access to the single market.
This could harm transactions in the City of London, hit cross border trade and stop British airlines from landing their planed in EU airports.
The hardline stance means Britain would have to swallow new EU laws without any right to challenge them, or face heavy punishments which would harm the economy.
Tory MP Jacob Rees-Mogg, head of the influential backbench group of Tory Brexiteers known as the European Research Group, hit out at the plan.
He told Mail Online: "The UK cannot accept ECJ jurisdiction if the EU itself will not.
"I wonder if such an approach would not violate the EU's own Charter of Fundamental Rights."
The plans reveal that the EU is worried that Britain will break European law during the transition period.
They fear that the UK will lodge an objection to the law with the ECJ and try to wait out the two years by embarking on a lengthy legal challenge at the Luxembourg based court.
They believe this will create a loophole which will effectively let Britain dodge new rules and regulations f or the transition period - something hey are determined to block.
To stop this happening Brussels wants to create a way of punishing Britain by denying important and lucrative sectors of the economy access to the single market if we do not follow all the new laws.
Brussels is insisting that Britain signs up to all EU laws and ECJ judgements during the two year transition period.
It states: "In addition, the Governance and Dispute Settlement Part of the Withdrawal Agreement should provide for a mechanism allowing the Union to suspend certain benefits deriving for the United Kingdom from participation in the internal market where it considers that referring the matter to the Court of Justice of the European Union would not bring in appropriate time the necessary remedies."
Peter Ptassek, Germany's top Brexit official, tweeted: "Now we are talking #transition! Straight forward draft by [the commission] provides the clarity the economy facing Brexit needs. Not so much time left to get a deal on this in March."
But the move will meet with fierce resistance among Tory Brexiteers who have warned against allowing Britain to become a "vassal state" during the two-year transition.
Brexit Secretary David Davis, the Brexit Secretary, has called for a clause to ensure that Britain will not be forced to implement laws against its interests.
But it looks as if that demand appears likely to be ignored by Brussels.
