President Obama demanded that Israel agree to a one-sided ceasefire during a heated phone conversation with Prime Minister Benjamin Netanyahu, according to a report aired on Israeli TV Tuesday.

Both leaders quickly denounced the purported transcript of their private discussion as fake.

But the TV channel that obtained it, state-run Channel 1, stood by its account.

In one bitter exchange, Obama declared that Qatar and Turkey — Hamas’ biggest international supporters — would mediate with the terrorists on Israel’s behalf, according to the TV channel.

“I trust Qatar and Turkey. Israel is not in the position that it can choose its mediators,” the president reportedly said in Sunday’s 35-minute call.

The White House and ­Netanyahu’s office angrily brand­ed the transcript bogus.

The journalist who broadcast the report, Oren Nahari, is a Channel 1 veteran who said he got the transcript from a “senior” US official and insisted it was authentic.

Nahari said the conversation began congenially, with Netanyahu thanking Obama for the administration’s support of Israel.

But things got tense when the talk turned to Israeli airstrikes on Gaza.

“I demand that Israel agrees to an immediate, unilateral ceasefire and halt all offensive activities, in particular airstrikes,” Obama reportedly said, prompting the prime minister to ask what Israel would get in exchange.

“I believe that Hamas will cease its rocket fire — silence will be met with silence,” Obama responded, according to the TV outlet.

Netanyahu reportedly answered that Hamas has broken five previous ceasefires, calling the group “a terrorist organization dedicated to the destruction of Israel.”

But Obama then reportedly added, “The pictures of destruction in Gaza distance the world from Israel’s position.

“Within a week of the end of Israel’s military activities, Qatar and Turkey will begin negotiations with Hamas based on the 2012 ­understandings, including Israel’s commitment to removing the siege restrictions on Gaza,” Obama added, according to the transcript.

“Qatar and Turkey are the biggest supporters of Hamas. It’s impossible to rely on them to be fair mediators,” the prime minister reportedly replied.

“The ball’s in Israel’s court, and it must end all its military activities,” Obama supposedly replied.

Washington and the prime minister’s officer issued statements saying the report was false.

“Shocking and disappointing someone would sink to misrepresenting a pvt convo between POTUS and PM in fabrications to Israeli press,” the US National Security Council tweeted.

Netanyahu’s office said, “We have seen these reports, and neither the reports nor the alleged transcript bear any resemblance to reality.”