"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", O'Neill said. "If others decide that they are not coming back into the devolved administration here in Northern Ireland then those issues will have to be dealt with at Westminster", she said.
The Prime Minister may not be present as the talks continue because she is heading to Paris for a meeting with newly elected president Emmanuel Macron.
Yesterday, Mrs Foster described her discussion with Mrs May as "very good", insisting she hopes to reach a conclusion sooner rather than later.
"If that's not possible, the Labour Party stands ready to offer strong and stable leadership in the national interest".
International Human Rights committees have repeatedly found Northern Ireland's abortion laws to be in breach of human rights standards and have called upon the UK Government to remedy this.
While the DUP backed Brexit, it has already stated that it opposes a hard border and wants to maintain the Common Travel Area which would allow the movement of goods and services to go unhindered.
The DUP has also made clear it does not believe leaving the EU should hurt the economy - potentially opening the door to customs union membership.
We know other matters, such as cutting air passenger duty and increased infrastructure spending, have been discussed, but we haven't got any sense of the exact details of the deal.
Mr Adams said his party will not take its seats in Westminster or swear an oath to the Queen, joking: "No harm to her".
Sir John said he feared that the United Kingdom government would no longer be seen as an "honest broker" between the sides, suggesting the "hard men" could decide to take matters into their own hands again. They supported Sinn Féin.
May's pursuit of the DUP has raised concerns in the Conservative Party.
The Tories and the DUP are considering a "confidence and supply" arrangement which would see the Northern Irish party back the Government to get its Budget through and on confidence motions.
But the other Stormont parties have warned that any deal between the DUP and Conservatives could make restoring power sharing more hard. These include a commitment by London to do more to attract Foreign Direct Investment into the province, improvement in connectivity between Northern Ireland and Great Britain - including a guarantee of direct access to London's hub airports.