Alexis Tsipras wrote to Ms Lagarde, warning that the IMF repayment would be missed unless the European Central Bank immediately raised its curbs on Greece's ability to issue short-term debt.
The letter, first reported by the Greek daily Kathimerini but independently confirmed by the Financial Times, raises questions about how close Athens is to bankruptcy. In addition to payments due to the IMF next month totalling €1.5bn, the Greek government has struggled to meet its wage and pension bills, which must be paid at the end of the month.
The next €300m IMF payment is due on June 5.
Greece is locked in a four-month stand-off with bailout lenders over the release of €7.2bn in aid that Athens needs to pay looming bills.
Creditors will not disburse the tranche without agreement on Greek economic reforms.
The contents of the Greek prime minister's letter were revealed by Ms Lagarde at a closed-door meeting of the fund's board on Thursday.
According to officials briefed on the talks, Poul Thomsen, head of the IMF's European department, warned the board that negotiations on the Greek economic reform package remained so unproductive that the fund could be forced to withhold its €3.6bn portion of the €7.2bn aid tranche.
Mr Thomsen insisted that the fund remained flexible on which reforms Athens needed to implement but told the board the programme must "add up" and begin lowering Greek debt to sustainable levels, officials said.
As a result, any watering down of Greece's reform package or lowering its budget surplus targets — positions advocated in some eurozone capitals, including Brussels — might require the EU to consider writedowns on its bailout loans in order for the IMF to assent, Mr Thomsen told the board.
Officials said Ms Lagarde fully backed Mr Thomsen, telling staff that they should not proceed with a "quick and dirty" approval process.
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"It's clear that we are very far from something the IMF will be able to support without fundamentally breaking its own rules," said one official briefed on the fund's board discussion.
According to two officials briefed on the talks, at least one board member raised the possibility of presenting a "take it or leave it proposal" to Greece.
However, IMF staff said they still did not have enough data from Greek authorities to put together such a plan.
A similar tactic was used in March 2013, when the Cypriot government was presented with a severe bailout plan and told it must agree or lose ECB support for its failing banking sector.
The idea of a "Cyprus-like" presentation to Greek authorities has gained traction among some eurozone finance ministers, according to one official involved in the talks.
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The official noted that the recent public backing by Wolfgang Schäuble, Germany's finance minister, for a Greek referendum fits into such a scheme. Under this scenario, Mr Tsipras would take the bailout ultimatum to a nationwide vote for approval.
However, another official involved in the talks cautioned that a "take it or leave it" approach remained only one of many ideas being discussed informally as a way to finalise an agreement.
Additional reporting by Kerin Hope in Athens