Foster has said that is part of her "blood red lines" over Brexit.
But he said a "Canada-plus-plus-plus" deal was "within our grasp with political will and imagination".
Conservative MP Steve Baker said on Tuesday that at least 40 of his colleagues are ready to vote against a deal which included that solution because it would represent a "half in, half out Brexit".
Speaking after meeting with the EU's chief negotiator Michel Barnier, she illustrated what the DUP were opposing by saying an attempt to avoid a hard border with the Irish Republic could mean more regulatory checks on goods travelling between GB and NI.
Theresa May has 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.
Mr Baker told BBC Radio 4's Today programme that he was expecting Tory Whips to go to work on the rebels - but that around 40 of them would stand firm against her.
The Conservatives will lose the next election if Theresa May doesn't change her Brexit policy, ex-cabinet minister David Davis has warned in a stinging letter to Tory MPs.
In a letter to fellow Tory MPs, he said a deal based on Mrs May's Chequers plan would deliver "none of the benefits of Brexit" and reduce the United Kingdom to being "a rule-taker from Brussels".
She told reporters: "There can not be barriers to trade in the United Kingdom internal market".
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.More news: George W. Bush's daughter Barbara gets married in secret ceremony
An announcement on what a possible deal could look like on the Irish border issue could be made as soon as Monday.
"The Future of England Study", conducted by YouGov on behalf of the Universities of Cardiff and Edinburgh, discovered that fully 87 percent of Ulster voters who backed Brexit believe an "unravelling of the peace process in Northern Ireland" would be a price worth paying to "take back control" from the European Union, according to the Belfast Telegraph.
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".
Repeating his call for Mrs May to ditch the plan agreed at her country residence in July, he said: "In the referendum both sides said Leave meant leaving the customs union and single market".
Responding to a question of whether she supported Mrs May's Brexit strategy, Ms Mordaunt replied: "The Prime Minister can count on my support but what I would say is we don't know where this is going to end up".
"We are awaiting the detail of exactly what we are going to be asked for vote for".
In a sign of progress, diplomats in Brussels told Reuters the European Union no longer expected a new proposal from Britain for the Ireland-UK border fix after Brexit as negotiators from both sides were seeking to agree on it in direct talks.
The de-facto Deputy Prime Minister was pressed on the DUP's support when he appeared on ITV's Peston on Wednesday night.
As part of the EU, Britain benefits from seamless trade with the bloc, its biggest trading partner.