BRUSSELS — European creditors issued Greece with an ultimatum Monday, saying the country must accept a key condition in bailout talks by the end of the week or face having to meet its debt commitments on its own – a prospect that many in the financial markets think would leave Greece little option but to leave the euro.

After a meeting of the 19 finance ministers of the eurozone over how to make Greece’s debts sustainable broke down in seeming acrimony after barely more than three hours, Greece was told it has to ask for an extension to its bailout program before further negotiations on the country’s future financing and economic course can take place.

“We simply need more time and the best way for that at this point is extend the current program which would allow a number of months for us to work on future arrangements,” said Jeroen Dijsselbloem, the head of the so-called eurogroup.

Without some sort of financing arrangements in place after the current bailout ends after Feb. 28, Greece would face real difficulties meeting its obligations, such as debt repayments, over the coming months. Bankruptcy and a potential exit from the euro would loom for Greece.

That’s why investors grew increasingly concerned Monday that a deal may not emerge in time to avoid a so-called “Grexit” from the euro – the main stock market in Greece fell 3.8 percent.

Investors are worried that the two sides are poles apart especially as a cornerstone of the election campaign of Greece’s new left-wing government was to scrap the bailout. In return for about $275 billion of rescue money, successive Greek governments have had to implement an array of austerity measures.