Stressing that her party wanted a deal that was "good for everyone" and "sustainable" in the long-term, Ms Foster said: "Good progress has been made and we will continue to work towards more progress".
Varadkar, who on Sunday warned May that time was running out for Britain to spell out exactly what kind of a post-Brexit deal it wants from the European Union, will hold a meeting with the British prime minister while the two leaders are in Belfast, his office said.
Sinn Fein's president Mary Lou McDonald (left) and Sinn Fein's vice president Michelle O'Neill, speaking to the media at Stormont Parliament buildings as Britain's Prime Minister Theresa May and Taoiseach Leo Varadkar are holding crunch talks at Stormont House.More news: Officers investigate suspicious white powder in Houses of Parliament
"We are very hopeful that those two parties are able to come to an agreement this week", he added.
She will also tell them that the UK Government is ready to introduce legislation to enable the re-establishment of an Executive as soon as possible following an agreement.
"Clearly, we need to meet [the DUP]", McDonald said.
The British province has been without a devolved executive for over a year since Irish nationalists Sinn Fein withdrew from a power-sharing government with their arch-rivals, the Democratic Unionist Party (DUP).More news: Qatar committed to supporting Iraq reconstruction: FM
Her statement continued: "I have made it consistently clear that unionists will not countenance a stand alone or free standing Irish Language Act". Under McDonald, Sinn Fein will continue to support close ties with the south that would include maintaining a so-called "soft border" with EU-member Ireland, with an eye on the party's core platform - unification.
'I believe in a woman's right to choose and that has been my position for a very, very long time, ' she said.
Mrs May and Taoiseach Leo Varadkar travelled to Belfast on Monday, amid speculation that a deal was within reach.
On the issue of Brexit and the Irish border, the premier said he sought and received assurances from Mrs May that the first stage agreement reached between the two governments in December would be put into law.More news: Fossil explodes higher by 80% after blockbuster earnings