Prime Abb Theresa May has urged Northern Ireland’s parties to make one final shoulder towards an agreement to restore devolved government.
Irish prime plenipotentiary Leo Varadkar said he was hopeful that a deal would be struck this week.
They were selected after a series of meetings with the parties in Northern Ireland.
The DUP and Sinn Féin clothed said that a deal is possible following Monday’s talks.
Speaking afterwards, DUP head Arlene Foster said: “There isn’t a deal yet, what there is, is quite good progress.”
Sinn Féin President Mary Lou McDonald believed the talks had reached “a decisive phase”.
Mrs May said that “it was now at the point where it’s stretch for local elected representatives to find a way to work together”.
She said she had “to the utmost and frank conversations with the main parties” and that while “some disagreements remain, I believe it is possible to see the basis of an agreement here”.
Taoiseach (Irish prime support) Leo Varadkar said he was hopeful that the two parties could “come to an pact this week”.
“The soundings from Arlene Foster and Mary Lou McDonald were confirming,” he added.
Northern Ireland has been run by civil servants since the power-sharing president made up of the DUP and Sinn Féin collapsed in January last year.
The then delegate first minister, Martin McGuinness, pulled Sinn Féin out of the coalition after a series of clashes with the final straw being the DUP’s handling of a scandal over non-professional energy scheme.
Mrs Foster said her party wanted a deal that was “sustainable into the expected”.
Ms McDonald said: “What is required now is the political will to prove to be it happen.”
SDLP leader Colum Eastwood called on the DUP and Sinn Féin to “visit messing about”.
“We still don’t know the colour of this deal,” he mean.
Prior to arriving at Stormont, Mrs May visited Bombardier’s plant in Belfast.
She met plant floor workers and was shown the capabilities of the multi-million pound plant which is the most important production site for wings for the C-Series.
Her visit came just to two weeks after Bombardier’s victory in a trade dispute with equal aerospace firm Boeing.
The US International Trade Commission (ITC) overturned a resolution to impose damaging tariffs on US imports of its C-Series aircraft, which intention have placed jobs in the Belfast factory at risk.
Several above-mentioned rounds of talks between the DUP and Sinn Féin have, so far, failed to wear out the political deadlock.
Earlier, a DUP source told the BBC a deal “is not likely” on Monday.
Mrs May in visited Northern Ireland in May when she toured the Balmoral Show vanguard of a snap general election.
Since then, she has struck a parliamentary trade with the DUP and has faced criticism for adopting a hands-off approach to the deadlock at Stormont.
Amid the issues that have divided the parties is Sinn Féin’s require for an Irish language act.
By BBC News NI political correspondent Mark Devenport
The Irish dialect act has been a major stumbling block, with DUP leader Arlene Succour promising previously that it wouldn’t happen on her watch.
What I am led to swear by is that there could possibly be a very strange, legislative fudge which would see the genesis of three separate bills, one for the Irish language, one dealing with Ulster Scots and one for wider cultural broadcasts.
The suggestion is that these three bills would end up merging together as one act in an essay to spare blushes on all sides.
On marriage equality, I am not exactly sure how they are prevailing to sort that one out, but one option that has been discussed is the possibility of leading a private member’s bill on the issue or deferring responsibility to Westminster.
At minor three or four DUP MLAs would be pretty hostile to the notion, but equally there are some who would succour it, or at least not object
A Downing Street spokesman said the prime agent would remind the parties of the many pressing issues facing Northern Ireland and descry clear her belief that a fully functioning executive is the best way to call the interests of the whole community.
“The (Irish) government has consistently said that the restoration of the sanatoria is essential in the context of full implementation of the Good Friday Agreement, and that it devise continue to work very closely with the British government to buttress the northern parties to achieve this outcome,” he said.
At the weekend, Sinn Féin’s infirmity president Michelle O’Neill predicted that the Stormont talks thinks fitting conclude this week.
Her comment strengthened speculation that a compromise between her dinner party and the DUP could soon be unveiled.