Jo Johnson, who had been a senior minister attending meetings of his brother's Cabinet, said it was an "unresolvable tension".
Setting the scene for a "people versus Parliament" election strategy, the spokesman said: "Boris will argue that it is now time for the people to decide after Parliament has failed them so we can resolve this once and for all".
Johnson's determination to lead Britain out of the European Union on October 31 come hell or high water is facing strong opposition from lawmakers, including members of his own Conservative Party who oppose to a no-deal Brexit.
The Labour leader had said the Bill must be passed through the Lords and have received Royal Assent before he would entertain the thought of heading to the polls.
Johnson remained determined to secure an election after lawmakers on Wednesday rejected his attempt to trigger a snap poll.
He was the MP for the Orpington constituency on the outskirts of London, and Minister of State for Universities, Science, Research and Innovation.
One Nation group leader and former Cabinet minister Damian Green told the BBC: "I'm afraid it does look as though somebody has decided that the moderate, progressive wing of the Conservative Party is not wanted on voyage".
"We have also received a commitment from the chief whip in the House of Commons that Commons consideration of any Lords amendments will take place on Monday and it is the Government's intention that the Bill be ready for Royal Assent", he told peers.
Johnson rose to power in July on a pledge to deliver Brexit next month - "deal or no deal" - and refuses to seek a delay.
He also acknowledged there were splits in the party about the timing of a general election, saying that the leadership was in contact with legal experts and other opposition parties about what to do. He told Parliament there was "real momentum" in negotiations with the EU, but European officials deny this and say Britain has yet to produce any concrete proposals. "So, we are now consulting about whether it's better to go long, therefore, rather than to go short".
The next meeting between European Union officials and the Johnson government's Brexit Sherpa, David Frost, will take place on Friday, but a Commission spokesperson said "concrete proposals" relating to alternatives to the Irish backstop, which Johnson has identified as the main obstacle to an agreement, had still not been received by the EU.
The prime minister is working to contain fallout from his decision last month to suspend Parliament until mid-October.
A cross-party alliance of MPs moved Tuesday to take control of the parliamentary agenda, setting the stage for another vote on Wednesday which could see MPs pass legislation that would delay Brexit until January 31 unless an exit deal is approved beforehand.
On Thursday, the Priem Minister hosts Israeli PM Benjamin Netanyahu and US Vice President Mike Pence in Downing Street.