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Johnson was compelled by a law, passed by opponents last month, to ask the bloc for an extension to the current Brexit deadline of October 31 until January 31 after MPs thwarted his attempt to pass his EU divorce deal on Saturday.
The government source said Johnson has now sent a total of three letters to Donald Tusk, the president of the European Council: a photocopy of the text that the law, known as the Benn Act, forced him to write; a cover note from Britain's EU envoy; and a third letter in which he said he did not want an extension.
"I have made clear since becoming Prime Minister and made clear to parliament again today, my view, and the Government's position, that a further extension would damage the interests of the UK and our EU partners, and the relationship between us," Johnson said in the third letter, published on Twitter by the Financial Times' Brussels correspondent.
Johnson said he was confident that the process of getting the Brexit legislation through Britain's parliament would be completed before October 31, according to the letter.
Tusk said he had received the request from Johnson.
"I will now start consulting EU leaders on how to react," he said on Twitter.
Johnson had hoped that Saturday would see recalcitrant lawmakers finally back the divorce deal he agreed with EU leaders this week and end three years of political deadlock since the 2016 referendum vote to leave the bloc.
Instead, lawmakers voted 322 to 306 in favour of an amendment that turned Johnson's planned finale on its head by obliging him to ask the EU for a delay, and increasing the opportunity for opponents to frustrate Brexit.
Johnson has previously promised that he would take the country out of the bloc on October 31, and would rather be "dead in a ditch" than ask for any extension without explaining how he would do this while also complying with the Benn Act.
Saturday's amendment, put forward by former Conservative cabinet minister Oliver Letwin, deflated Johnson's big Brexit day just as hundreds of thousands gathered to march on parliament demanding another referendum on EU membership.
After several hours of heated debate, senior politicians - including Business Secretary Andrea Leadsom, House of Commons leader Jacob Rees-Mogg and Labour's foreign affairs spokeswoman Diane Abbott - were escorted from parliament past jeering demonstrators by phalanxes of police.
Jeremy Corbyn, leader of the main opposition Labour Party, backed a second referendum, saying "the people should have the final say".
The European Commission said Britain must now inform it of its next steps as soon as possible.
French President Emmanuel Macron told Johnson a delay was in no-one's interest, an official at the French presidency told Reuters.
Ireland believes granting an extension is preferable to Britain leaving with no deal, but there is no guarantee that view is shared throughout the EU, its foreign minister said.
Brexit "Super Saturday" topped a frenetic week which saw Johnson confound his opponents by clinching a new Brexit deal with the EU.
When it comes to a vote in a divided parliament where he has no majority, Johnson must win the support of 320 lawmakers to pass his deal.
If he wins, he will go down in history as the leader who delivered a Brexit - for good or bad - that pulls the United Kingdom far out of the EU's orbit.
Should he fail, Johnson will face the humiliation of Brexit unravelling after repeatedly promising that he would get it done - "do or die" - by October 31.
Johnson's predecessor Theresa May was forced to delay the departure date. Parliament rejected her deal three times, by margins of between 58 and 230 votes, earlier this year.
He says lawmakers face the option of either approving the deal or propelling the United Kingdom to a disorderly no-deal exit that could divide the West, hurt global growth and bring renewed violence to Northern Ireland.
To win, Johnson must persuade enough Brexit-supporting rebels in both his Conservative Party and the Labour Party to back his deal. His Northern Irish allies and the three main opposition parties oppose it.
Some influential hardline Brexit supporters have said they will support the deal.