London (CNN) It's been a turbulent couple of months for Britain's beleaguered Prime Minister Theresa May, whose political survival has been an unrelenting source of speculation.

With June's snap election failing spectacularly to deliver May's stated desire for a stronger governing mandate, she has increasingly steered away from her intransigent approach to Brexit to a more pragmatic campaign of negotiations.

But it will be talks in this week that could have far-reaching implications, not just for May, but the nation as well.

Deadline looms

On Monday, the British Prime Minister will head to Brussels to meet with European leaders Jean-Claude Juncker and Donald Tusk.

Negotiations between the European Union and the UK thus far have been frustratingly slow . May will be hoping the EU heads of government determine "sufficient progress" has been made and give the green light for future trade talks at a critical EU summit on December 14.

Ten days ago, the British PM was given a deadline of December 4 to put forward further proposals over three key sticking points: the Northern Ireland border , Britain's financial settlement and the rights of EU citizens

The so-called "divorce bill" -- the money May's government must pay into the EU budget as part of its current membership obligations -- has reportedly been resolved. But if the two sides can't come to an agreement on what kind of border will run between (British) Northern Ireland and the (EU) Republic of Ireland, the mid-December summit might not produce a breakthrough.

Britain has said it will leave both the single market and customs union when it leaves the bloc in March 2019 -- a move that critics say could lead to a so-called "hard border" in Ireland.

Tusk: Ireland has final say

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Tusk, the European Council President, was adamant Friday that the Ireland issue must be resolved before any post-Brexit trade discussions can move forward.

"If the UK offer is unacceptable for Ireland, it will also be unacceptable for the EU," Tusk told reporters in Dublin. "This is why the key to the UK's future lies -- in some ways -- in Dublin, at least as long as Brexit negotiations continue."

Ireland has said it is not looking to delay the Brexit process but wants written assurances in the form of a specific and detailed border plan.

Irish Foreign Minister Simon Coveney said Sunday he was unsure if an agreement dodging a hard Northern Ireland border could be managed by Monday's deadline, but he hoped that May's meetings in Brussels might lead to a rapprochement.

"Let's not run before we can walk here. Obviously, we would like that to be the case," Coveney told RTE radio, according to Reuters

"The hope is that those (Monday) meetings will result in a momentum that can be carried into the leaders' summit the week after ... and can allow this Brexit negotiation process to open up to phase two of discussions."

And May also faces potential obstacles from Northern Ireland's Democratic Party, whose 10 Westminster lawmakers are propping up her minority government and who will oppose any special treatment for Northern Ireland.

New Brexiteer demands

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Meanwhile back at home, top Brexit supporters have demanded that May stand her ground on "any further financial commitment to the EU until they have agreed that in return, they will meet a number of conditions."

Preempting the mid-December summit, the Leave Means Leave group sent a letter outlining several terms, including calling for Britain to be beyond the jurisdiction of the European Court of Justice (ECJ) and for no new EU regulations to apply once Britain exits in March 2019.

Former Conservative Party leader Iain Duncan Smith wrote in The Sunday Telegraph that the Brexit divorce bill should only be paid once the ECJ has relinquished control.

"It must be contingent on an end to the authority of the European Court of Justice. Most people who take an interest in the EU recognize that the single most important definition of taking back control is the moment we leave the authority of the ECJ," Duncan Smith wrote.

"Yet even as we get ready to leave, the EU has insisted that the ECJ retains the right to rule on the rights of EU citizens in the UK after Brexit ... To do this, they demanded the jurisdiction of the ECJ should be maintained. This is the most preposterous claim."

While May has promised to end the ECJ's authority in the UK, she has alluded that its remit might continue in some capacity during an "implementation period" past March 2019.

This will not placate the Leave Means Leave camp, and could cause problems at home for the embattled PM.

The coming week brings the negotiations to date to a climax, with disagreements potentially halting the entire process. The EU must deem there to be "sufficient progress" in all three areas of contention for Brexit talks to move onto the next phase in December.

How May handles the meetings on Monday and the upcoming summit will be crucial both in retaining her position as prime minister as well as indicating what a post-Brexit Britain may look like.