"One thing is clear: The place of Greece is and remains in Europe," said European Commission Vice-President Valdis Dombrovskis at the Commission's daily briefing in Brussels.

And Jeroen Dijsselbloem, the Dutch finance minister who presides over the Eurogroup bringing together finance ministers of all 19 eurozone members, said in The Hague that it was still his goal to keep Greece in the common currency.

Comments like these appeared to be taking a step back from EU rhetoric before Sunday's referendum, when high-ranked officials from Jean-Claude Juncker, the Comission's president, to Martin Schulz, president of the European Parliament, insisted a 'no' in the referendum would mean 'no' to Europe.

Italian Prime Minister Matteo Renzi said the Greeks were in fact choosing between the euro and the drachma.

'All problems can be solved'

On Monday, Renzi sounded more conciliatory, writing in a Facebook post that Italy would do its part to "resolve this emergency."

"None of us [eurozone finance ministers] wants Greece to leave the euro," Spanish finance minister Luis de Guindos told the Spanish media at a press conference in Madrid.

For his part, Finnish Finance Minister Alexander Stubb even gave off an air of optimism that a solution to the Greek debt crisis could be found when the eurozone's finance ministers converge in Brussels for yet another Eurogroup meeting.

This optimism, however, was not reflected in comments from the European Commission or the Eurogroup.

""When proposals are declined, matters become more complicated," Jeroen Dijsselbloem told the Dutch press.

"The 'no' result widens the gap between Greece and other eurozone countries," said Valdis Dombrovskis.

He added that while the commission was ready to continue working with Greece, there were currently no negotiations on either the political or the technical level on a possible agreement on further financial aid to Greece.

While Greek Prime Minister Alexis Tsipras had already requested a new bail-out from the eurozone's bailout fund called the European Stability Mechanism or ESM, the Eurogroup had unanimously declined to discuss this proposal before the referendum.

German Finance Minister Wolfgang Schäuble said on Monday he was waiting for his Greek colleague to present a new proposal when the Eurogroup meets on Tuesday in Brussels.

Following the resignation of the controversial Yanis Varoufakis as finance minister, the former chief negotiator in bailout talks, Euclid Tsakalatos, will represent the Greek government as the new finance minister.

Greece in dire financial straits

Whatever a new proposal from Greece contains, it will need a fast decision as Greece's financial and economic situation is deteriorating quickly.

After negotiations had been dragging on for months, a second bail-out program expired last week on June 30, 2015, leaving Greece without financial assistance for the first time since the crisis began in May 2010.

Given its inability to tap into the remaining assets of the second bail-out, Greece has had to introduce capital controls limiting withdrawals of domestic card holders to 60 euros per day. Banks have been closed since last Monday , but were scheduled to open again this Tuesday, July 7. Reports suggest they will now stay closed until Wednesday.

Last Tuesday also saw Greece's defaulting on a roughly 1.6-billion-euro loan by the International Monetary Fund (IMF).

European Central Bank holding lifeline

Currently, it is the European Central Bank (ECB) that is holding the lifeline for Greek banks with "emergency liquidity assistance" or ELA. While the ECB has continuously increased this emergency cash, it has kept it at the level of 89 billion euros since the referendum was announced.

Assessments of how fast Greek banks would run out of cash once banks open again differ widely, with predictions ranging from a few days to several weeks.

"The use of capital controls gives Greek banks more than a month of liquidity, even if the ECB does not increase emergency liquidity assistance," Diego Valiante of the Center for European Policy Studies told Deutsche Welle.

Debt one divisive issue

Any new Greek proposal is likely to include the request for debt relief, or at least a restructuring of Greece's debt.

Greece has requested a so-called haircut for many months, and the government appeared to find vindication in a recent IMF report finding that Greece's public finances were unsustainable and a haircut would be necessary.

For other eurozone countries, a haircut has been out of the question.

"Spain also has very high debt," explains Pablo Rodriguez of Spain's El Mundo Newspaper. "If Greece got its haircut, the left-wing 'Podemos' party in Spain will try to exploit that for its own campaign be very happy to campaign on getting debt relief for Spain as well in the upcoming elections."

But Daniel Gros of the Center for European Policy Studies says a final agreement could include something that would in fact amount to a haircut even though it would not be called by that name.

"In political terms, the question is whether [in the upcoming meetings] a formula is found that Alexis Tsipras can use at home to say he achieved a haircut, and that at the same time German finance minister Wolfgang Schäuble can take back and say that there was no haircut."