The BBC and Sky, without citing sources, reported that the DUP could vote against the budget on October 29 if a deal meant new barriers between the British province and the rest of the United Kingdom.
But she appeared to reinforce the party's red line on checks along the Irish Sea in the event of the backstop taking effect.
The new plan has not yet been formally tabled but details which emerged last week sparked a wave of optimism that a Brexit can be reached.
Barnier said Brexit would trigger the need for customs, Value-Added Tax and compliance checks with EU standards between Ireland and Northern Ireland in the event that a planned "backstop" were triggered because a future EU-UK trade deal was not sufficient in itself to ensure the land frontier was not a "hard border".
"We need to be satisfied that we're not going to do damage to the United Kingdom ..."
Weakened by both the absence of an Assembly - which means she has now not been first minister for nearly two years - and a slew of devastating revelations at the ongoing public inquiry into the "cash for ash" scandal, Mrs Foster is in a particularly weak position within the DUP from which to compromise, even if she was willing to countenance wavering on her Brexit red line.
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"There can not be any customs or regulatory barriers between ourselves and the rest of the United Kingdom, both ways", said Mrs Foster following talks with European Union chief negotiator Michel Barnier in Brussels. "Such a customs union would eliminate an important part of customs checks".
The British government and leading politicians in Northern Ireland insist they will not agree to a withdrawal treaty that foresees the possibility of Northern Ireland being kept inside the EU's economic area while the British mainland is not.
"That's why the backstop proposed by Barnier is unacceptable to us". Asked whether that implied her supporters are ready to "fight and die" for them, she replied: "What I meant by that is that they are deep red - they're not pink".
But he made clear that the European Union envisages a deal involving new customs and regulatory checks on goods travelling from mainland Britain to Northern Ireland, including health and sanitary inspections for all animals and animal products arriving in ports like Larne and Belfast from the rest of the UK.
The DUP's Brexit spokesman, writing in the Daily Telegraph, said his party would not accept any deal that would leave Northern Ireland "languishing in the stifling embrace of the EU".
Foster stressed that she wants to reach a Brexit deal and she is staying in Brussels for the next two days in order to try and find a solution that would be acceptable to her.