REUTERS/Alkis Konstantinidis Greek Prime Minister Alexis Tsipras looks at his watch before addressing lawmakers during a parliamentary session for the creation of a committee for claiming World War II war reparations at the parliament building in Athens March 10, 2015. 
Greece is due to make a €305 million ($339.07 million, £218.64 million) disbursement on June 5, followed by three others in the following two weeks.
Greece had to use its special drawing rights (SDRs) held at the IMF to make the last payment. - the four coming up in June are twice as large as that one, and come after another round of pension and salary payments.
Parliamentary speaker and Syriza politician Nikos Filis made the admission on TV, according to Reuters:
"Now is the moment that negotiations are coming to a head. Now is the moment of truth, on June 5," parliamentary speaker Nikos Filis, from the ruling Syriza party, told ANT1 television.
"If there is no deal by then that will address the current funding problem, they won't get any money," he said.
France's La Tribune is also reporting that the government now knows it won't be able to make the June 5 payment, calling that the new "critical date" by which time a deal must be reached. If it's not managed by then, Greece will default.
The government's spokesman didn't seem to be too worried about the possibility:
#Syriza spox Petrakos replies: "What can happen to us [if we don't pay the IMF]? Nothing is going to happen to us!" #Greece
- The Greek Analyst (@GreekAnalyst) May 20, 2015That's likely to be just bluster, since any failure to make contracted debt repayments could result in the Greek banking sector being cut off by the European Central Bank (ECB).
Athens has seemed more confident that a deal is coming in recent days. An agreement would unlock billions of euros for the cash-strapped government, and allow them to cover many of their upcoming debt payments, at least for a few months. This week, a leak to Greek newspaper To Vima suggested that the European Commission had offered a deal, the terms of which sounded much less stringent than expected.
Quick after, finance minister Yanis Varoufakis said that Greece will have a deal within a week.
That doesn't sit well with what we heard from the IMF over the weekend - with a leaked memo suggesting that though progress had been made on some issues, there were still major areas of contention. If the negotiators can bridge those gaps in two weeks, then Greece will get more than enough cash to make the upcoming IMF payments. If it can't, then then the crunch is coming.
Here's the whole timeline of payments coming up, courtesy of HSBC:
HSBC, Bloomberg, IMF, Business Insider