"I will be making it very clear that any deal between the Tories and the DUP can not be allowed to undermine the Good Friday and subsequent agreements", Ms O'Neill said in a statement.
The BBC reported that a deal with the DUP was expected to be signed on Wednesday.
It had been suggested the start of negotiations could be delayed by the failure of any party to win a House of Commons majority at last week's General Election.
Sky Sources say that "significant progress" has been made today, while Mrs Foster tweeted to say she hopes to bring a successful conclusion soon.
Scottish Conservative leader Ruth Davidson said the government should put economic growth at the heart of its Brexit strategy, comments placing her in the camp of those advocating a closer trade relationship with the European Union, or "soft" Brexit.
Meanwhile, talks are continuing between the DUP and Conservatives, to secure the support of the DUP's 10 MPs in steering government business, including crucial measures on Brexit, through the Commons.
The first round of talks are likely to focus on the UK's so-called divorce bill, estimated to be as much as €100bn (£88bn), as well as the rights of European Union citizens now living in Britain and United Kingdom nationals living on the continent.
Some observers have maintained that a deal with the DUP risks destabilising Northern Ireland by increasing the influence of pro-British unionists.
But Jeremy Corbyn, leader of Britain's main opposition, said his Labour Party would not support May's Queen's Speech in the lower house of parliament to try to force her out of power through a vote of no confidence. Devolved government at Stormont, site of the Northern Ireland assembly, broke down in January and has yet to restart.
The EU will keep the door open for Britain to return, but only on worse terms than it now has, European Parliament Brexit negotiator Guy Verhofstadt said Wednesday. "It continues and when the deal is done, it will be done", a senior source in the Conservative Party said on Thursday.
The DUP said it was discussing issues to do with Brexit and economic matters in its talks with the Conservatives.
Britain's Prime Minister Theresa May and her husband Phillip arrive at church in Sonning, Britain.
The Times newspaper said finance minister Philip Hammond would push May not to leave the customs union - an arrangement which guarantees tariff-free trade within the bloc but prohibits members from striking third-party trade deals.
Sinn Fein said that it was important there was transparency over any deal between the Conservatives and the DUP, and that compromises over certain policy areas could hinder efforts to form a new Northern Irish executive.
The talks were "productive", May told a news conference in Paris following a meeting with French President Emmanuel Macron.
She has since promised to set up a public inquiry and pledged £5 million (S$8.83 million) of support and housing guarantees. A resident in a nearby building watches smoke rise from a building on fire in London , Wednesday, June 14, 2017.
He said one machinery room and two berthing areas for 116 crew members were severely damaged from the impact to the ship's side. Earlier, the 7th Fleet said the US ship sustained damage on its starboard side and experienced flooding in some spaces.
London police said Saturday that the death toll from the inferno at the Grenfell Tower in west London had risen to 58. He said the situation has become an emergency and that "all assets" should be brought to the table.
Trump has furiously railed against leaks coming out of his administration and the intelligence community since he took office. He has sometimes, without prompting, injected "I'm not under investigation" into conversations with associates and allies.
The trial was the first meant to replicate the flight profile of an ICBM that could be deployed by North Korea. US President Donald Trump had also warned that military intervention "was an option under consideration".
Republicans and Democrats sat in different parts of the park - fans could state their party preference when buying tickets. They sat next to each other at the game above the first-base dugout, as a show of friendship across the aisle.
Still, the snowballing investigation has deeply angered Trump, who denies he has any nefarious ties to Russian Federation . But Rosenstein, too, may ultimately have to hand off oversight given his role in Trump's decision to fire Comey.