BRUSSELS—As the bailout standoff between Athens and its creditors escalates, some European officials are suggesting something that was once unthinkable: Let Greece keep the euro currency even if it defaults on its rescue loans.

This idea breaks with the conventional wisdom of more than five years of debt crisis, where the shock of a default has been seen as sending Greece down an inexorable path of bank runs, capital controls and, finally, exit from the eurozone.

Yet, with the risk of nonpayment higher than ever, finding a way to avoid the chaos of a Greek currency switch is looking more attractive. Proponents say it could spare Europe the embarrassment of one of its members crashing out of the eurozone, damp some of the market panic that would likely follow a default, and avoid setting a precedent that would undermine confidence in those still inside.

The idea of keeping Greece in the euro despite a default was briefly discussed by senior officials from eurozone finance ministries last week, although many of them harbor serious doubts about if it would work, according to people familiar with the talks.

“It is not so much a plan, but an evolution in the thinking,” says one person familiar with the discussions among Greece’s creditors.

Proponents of a default-without-exit scenario largely fall into two camps: those who believe the shock of a temporary default would compel Prime Minister Alexis Tsipras to finally seal a financing deal with the creditors; and those who believe that an immediate ejection from the euro would trigger chaos in Greece and beyond.

“Greece lacks the capacity for launching a new currency and [organizing a] ‘Grexit,’ ” an official familiar with last week’s discussion said, using a popular term for describing Greece’s departure from the eurozone.

Any scenario where Greece fails to secure new funds from its international creditors would likely see the government issue a sort of parallel currency to pay wages and government contractors for some time, even as it keeps the euro as its legal tender, experts say.

“It’s the simple answer when you run out of cash,” says Harold James, a professor at Princeton who specializes in Europe’s financial history.

Parallel currencies have been around for centuries. In the late Middle Ages, merchants in Florence and the Netherlands paid local laborers and suppliers in silver coins while settling bigger transactions in gold—without a fixed exchange rate between the two.

In a world where foreign-exchange trades are conducted in split seconds, managing two separate currencies would pose more challenges.

Like in California, which issued IOUs in 2009 when a budget impasse left it unable to pay tax refunds, vendors and local governments, Greece’s parallel currency would likely take the form of debt issued to its own citizens.

But the Greek government would face immediate doubts over whether its new currency, or IOUs, would ever be converted into euros. That would make it less like California and more like Argentina, which defaulted on its sovereign debt in 2002.

Public entities facing budget problems issued a variety of IOUs, which sank below face value because of doubts about their creditworthiness and whether the exchange rate to the dollar would hold. It didn’t.

Complicating matters further, Athens would also depend on the European Central Bank—one of the first creditors likely to be jilted by a default—to provide at least some emergency funding to help Greek banks survive accelerating deposit outflows.

This kind of uncertainty would likely lead to an immediate depreciation of the new currency against the euro, with a black market in which physical euros would trade at a much higher price than that set up the government.

If the government at the same time resorts to capital controls and stops depositors from withdrawing their euro savings, a third exchange rate might emerge, with different prices for notes and coins and euros held in bank deposits. In contrast to Cyprus, which introduced capital controls in 2013 when its two biggest banks went bust, Greece wouldn’t have international bailout funds to help buffer the hit of severely constrained liquidity on its economy.

“It’s very, very disruptive. People will not do transactions that they would usually have done. So economic activity would drop off quite quickly,” Mr. James says.

Foreign companies might also be skeptical of doing business in the new currency, leading to shortages of products like medicine or spare parts for cars and machinery.

Perhaps most important, keeping the parallel currency credible would force the Greek government to do the very thing it has been fighting against: cutting government spending. “The only way it would really work is if it comes with a strict fiscal program,” Mr. James says.

That is why most economists and many policy makers believe that even if Greece were to go down that path it would eventually have to give up on the euro and the parallel currency would take over. In economics, the phenomenon is described as Gresham’s law: “Bad money drives out good.”

“You will not get the benefit from exiting the euro and having your own monetary policy,” says an official involved in the negotiations on Greece’s finances.