What is in Boris Johnson's Brexit plan?
What are the Brexit backstop options?
Both leaders said they could see a "pathway to a possible deal" after that meeting.
There "positive" comments after the meeting by both Mr Johnson and Mr Varadkar prompted speculation of compromises on customs arrangements to avoid border checks on the island of Ireland.
by BBC News NI political editor Mark Devenport
Details of the latest ideas being considered by London and Brussels haven't yet been clarified.
But it's thought they might involve an overlapping structure which could see EU customs tariffs being collected on goods moving from Great Britain across the Irish Sea.
However, rebates would be paid if those goods are intended for use only within the UK.
The DUP have said it won't support any arrangements unless it believes them to be in Northern Ireland's long-term economic and constitutional interests.
In what looks like a shot across Boris Johnson's bows, the party emphasises the pivotal role their ten MPs play at Westminster and point out that parliamentary arithmetic has not changed regardless of the ups and downs of the Brexit discussions.
In a statement on Friday, Mrs Foster said the party, whose support will be key to getting a Brexit deal through Parliament, would only back a deal that is in Northern Ireland's "long-term economic and constitutional interests".
She added: "We have been consistent in our opposition to the backstop, whether UK or NI only.
"Anything that traps Northern Ireland in the European Union, whether single market or customs union, as the rest of the United Kingdom leaves will not have our support - the prime minister is very mindful of that."
Mrs Foster said the party has "argued that it is important to secure a balanced and sensible deal as we leave the European Union".
"Those who know anything about Northern Ireland will appreciate that these issues will only work with the support of the unionist as well as the nationalist community."
She added that the DUP is "very relevant in the parliamentary arithmetic and regardless of the ups and downs of the Brexit discussions that has not changed".
Mrs Foster's comments on Northern Ireland's consent in post-Brexit arrangements come after NI secretary Julian Smith said there will not be a situation where "one community has a veto" over Brexit plans.
Speaking on BBC's The View on Thursday, Mr Smith was responding to concern from some NI parties that the prime minister's Brexit proposals could give the DUP a veto on post-Brexit arrangements.
Meanwhile, on Friday, Mr Smith confirmed on Twitter that he was having "positive discussions" with the main Northern Irish political parties during meetings at