Boris Johnson has said he would “rather be dead in a ditch” than ask the EU to delay Brexit beyond 31 October.
But the prime minister declined to say whether he would resign if this had to happen.
Mr Johnson has said he would be prepared to leave the EU without a deal, but Labour says stopping a no-deal Brexit is its priority.
The prime minister’s younger brother, Jo Johnson, announced earlier that he was standing down as a minister and MP.
Speaking in West Yorkshire, Boris Johnson said Jo Johnson was a “fantastic guy” but they had had “differences” over the EU.
He thanked his brother “for all the work he has done and the support he has given for our domestic agenda”.
Announcing his resignation earlier in the day, Jo Johnson said he had been “torn between family loyalty and the national interest”.
During his speech in Wakefield, the prime minister reiterated his call for an election, which he wants to take place on 15 October.
He argued it was “the only way to get this thing [Brexit] moving”.
“We either go forward with our plan to get a deal, take the country out on 31 October which we can or else somebody else should be allowed to see if they can keep us in beyond 31 October,” Mr Johnson said.
He told the audience he hated “banging on about Brexit” but accused MPs of having “torpedoed” the UK’s negotiating position with the EU by voting for a bill designed to block a no-deal exit on 31 October.
However, Labour leader Jeremy Corbyn has accused the PM of having “no plan to get a new deal”.
The government has announced that MPs will get another chance to vote for an early election on Monday.
This comes after the House of Commons rejected Mr Johnson’s plan for this in a vote on Wednesday.
The fresh vote on an early election is scheduled just before Parliament is due to be prorogued – or suspended – from next week until 14 October.
Announcing the vote, Commons leader Jacob Rees-Mogg said the suspension would begin on Monday, Tuesday or Wednesday but did not say precisely when.
Labour and other opposition parties say they will not back the prime minister’s call to have a general election while the option of a no-deal Brexit on 31 October remains open to Mr Johnson.