The MP dismissed suggestions that a Canada-style free trade deal would lead to a hard border being created between Northern Ireland and Ireland as a "red herring".
Avramopoulos also said the bloc's no-deal contingency plans on security were driven by "goodwill", stressing the bloc would want to continue cooperating with Britain closely, including on sharing intelligence.
The radical move is one of the options being considered by the DUP if attempts to nail down a deal with Brussels include any proposals that would leave Northern Ireland being treated differently to the rest of the United Kingdom, the Press Association understands.
Mrs May, meanwhile, urged MPs across the Commons to act in the national interest and back a Brexit deal amid warnings that "decisive" progress is needed in the negotiations before a crunch Brussels summit next week.
But she suggested that the Prime Minister may have to amend her Chequers plan to get a final deal, saying that "we don't know where this is going to end up".
Meanwhile, Mrs May told her Cabinet that Britain will not accept an European Union withdrawal deal without a "precise" political declaration setting out how its requirements on trade and security will be delivered.
'My estimate is that there are at least 40 colleagues who are not going to accept a half-in, half-out Chequers deal or indeed a backstop that leaves us in the internal market and the Customs Union, come what may'.
The key question is how to let Britain exit the European Union without too much disruption to trade and without reinstalling a hard border between the Republic of Ireland and Britain's Northern Ireland.
It would damage the economic well being of Northern Ireland.
"I hope for progress and there is progress, but the devil is in the details, as they say", said Merkel, who was in The Hague for talks with Dutch Prime Minister Mark Rutte.
But on actually getting a deal, he said: "We've always been clear we would aim for the October council but there would be leeway, that it might slip into November.
But since we can't guarantee the result right now, we have to prepare for all possible outcomes", one diplomat said.
The same group of MPs are also thought to be against the leadership's strategy of rejecting the deal the Prime Minister negotiates in order to trigger a general election.
Other cabinet members, who appeared to not have been invited to the meeting, including Esther McVey, the work and pensions secretary, and Penny Mordaunt, the global development secretary, were said to be frustrated with the lack of a time limit, and, according to The Times, citing an unnamed source, were considering their positions.
"There cannot be barriers to trade in the United Kingdom internal market which would damage the economic well-being of Northern Ireland and therefore we could not support any arrangements which would give rise to either customs or regulatory barriers within the United Kingdom internal market".
The party's Brexit spokesman Sammy Wilson later explicitly warned the Government against doing a deal that kept Northern Ireland in the Single Market, telling the Telegraph it could not support "any deal which includes such economically and constitutionally damaging arrangements".