Finland's Prime Minister, Antti Rinne, gave the ultimatum after a meeting with French President Emmanuel Macron.
"If the UK wants to discuss alternatives to the existing Brexit agreement then these must be presented before the end of the month," said Mr Rinne, according to reports from Finland.
He continued that he felt European leaders were in agreement that should this not be the case talks would be "over".
It comes after the PM held talks with European leaders on Wednesday, including European Commission President Jean-Claude Juncker.
A Downing Street spokeswoman said: "This afternoon the Prime Minister spoke to European Commission President Juncker, European Parliament President Sassoli, Cypriot President Anastasiades, and Latvian Prime Minister Karins.
"The Prime Minister and President Juncker discussed the positive and constructive conversation they had in Luxembourg on Monday and their shared determination to reach a deal."
Previously, Mr Juncker said the "risk of a no deal is palpable".
EU chief negotiator Michel Barnier also voiced the frustration of EU leaders, saying: “Some three years after the Brexit referendum we should not be pretending to negotiate.”
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