Theresa May will not be asking the EU for a long delay when she formally requests that Brexit is postponed.
Downing Street said the PM shared the public's "frustration" at Parliament's "failure to take a decision".
BBC assistant political editor Norman Smith said the delay would not be beyond the end of June.
A cabinet minister has told the BBC this would be the "wrong choice" and a "craven surrender to hardliners" within the Conservative Party.
Under current law, the UK will leave the EU - with or without a deal - in nine days.
The PM is due to send a letter requesting a delay to Brexit later, ahead of a EU summit on Thursday at which she will discuss the matter with fellow leaders.
Any delay will have to be agreed by all 27 EU member states and EU Brexit negotiator Michel Barnier has said the EU will not grant it without a "concrete plan" from the UK about what they would do with it.
Explaining that Mrs May "won't be asking for a long extension" when she writes to the EU, Number 10 said: "There is a case for giving Parliament a bit more time to agree a way forward, but the people of this country have been waiting nearly three years now.
"They are fed up with Parliament's failure to take a decision and the PM shares their frustration."
European Commission President Jean-Claude Juncker indicated there could be an extra Brexit summit next week.
But a tweet from his spokeswoman said his patience was "wearing thin" and that the withdrawal agreement would not be negotiated.
It comes after MPs rejected the withdrawal deal Mrs May has negotiated with the EU for a second time last week by 149 votes. They also voted in favour of ruling out leaving the EU without a deal, and in favour of extending the Brexit process.
The prime minister had hoped to have a third attempt at getting MPs to back her deal - but Speaker John Bercow effectively torpedoed that with his surprise intervention.
He said a third "meaningful vote" could not happen in the coming days if it was "substantially the same" motion.
Education Secretary Damian Hinds told the BBC that MPs had to "get on with" approving a Brexit deal - although he said another vote would not take place until ministers had "some confidence" of victory.
"You can't keep on kicking the ball further and further down the street," he told Radio 4's Today. "You have to pick up and run with it."
But an unnamed cabinet colleague told the BBC that asking for such a short delay was "weak, weak, weak".
"This substantially increases the risk of no deal," they said. "Her most craven surrender to the hardliners yet. She knows this is the wrong choice for the country but she's putting her short term interests first."
While Mrs May still wants to put her deal before MPs one more time, she has said that even if that goes ahead and they vote in favour of it, the UK would need a short extension to get the necessary legislation through Parliament.
Mrs May had warned Brexiteer Tories that a longer extension would be needed if her deal did not get through Parliament.
Norman Smith said that if the EU agrees to a three-month delay, it opens up the possibility that the third meaningful vote could take place after next week.
But he added that the risk of taking a long delay off the table is that "Brexiteers can now just sit on their hands until 30 June".
Source: BBC