More Labour MPs could quit the party unless it listens to their concerns, Jeremy Corbyn has been warned.
Seven MPs have walked out in protest at the Labour leader's handling of anti-Semitism and Brexit.
One of the seven, Chuka Umunna, said "a lot of Labour MPs" could follow suit, together with Tories "demoralised by the UKIPisation of their party".
Deputy Labour leader Tom Watson has warned his party could see more defections if it did not change.
He said Labour had to do more to tackle anti-Semitism and he also urged Mr Corbyn to reshuffle his shadow cabinet to reflect a wider range of MPs.
Mr Corbyn has said in a statement he was "disappointed" by the defections, which represent the biggest split in the Labour Party since the Social Democratic Party was set up 40 years ago.
Shadow Chancellor John McDonnell said the seven should now stand down as MPs and seek re-election against Labour Party candidates.
The seven MPs - Mr Umunna, Luciana Berger, Chris Leslie, Angela Smith, Mike Gapes, Gavin Shuker and Ann Coffey - quit Labour in protest at what they said was a culture of "bullying and bigotry" in the party and frustration over the leadership's reluctance to back another EU referendum.
Mr Umunna said another "big issue for us" was the belief that Mr Corbyn could not be trusted with national security, if he became prime minister.
"Many Labour MPs agree with us on that," he told BBC Radio 4's Today programme.
Mr Umunna said the new Independent Group was not yet a new political party, but he believed it could become one in time.
He urged members of all parties to join them in building an "alternative" to the current two party system, which he said was "fundamentally broken".
The BBC has been told two Conservative MPs are thinking of joining the new Independent Group in Parliament.
Mr Umunna refused to speculate on who they could be but he added: "There are a lot of Labour MPs wrestling with their conscience on this issue but also Conservatives who have become demoralised by the UKIPisation, if you like, of the Conservative Party."
A number of Conservative MPs are at the centre of rumours about joining the new group.
Sarah Wollaston, a supporter of the People's Vote campaign for another EU referendum, along with the seven Labour defectors, has warned about former UKIP members joining local Tory parties and the pro-Brexit European Research Group (ERG) pushing the party to the right.
On Monday evening, she tweeted: "#BLUKIP has been busy taking over the Tory Party alongside the ERG. Soon there will be nothing left at all to appeal to moderate centre ground voters."
Other Conservative MPs unhappy with the party's direction include Anna Soubry, another People's Vote supporter, who has called in the past for a new centre party.
A new political landscape?
This splintering might, just might - in time - turn into a much bigger redrawing of the landscape.
For now though that is way off. And this is first and foremost about the Labour Party - the seeds of the splinter sown more than three years ago, bearing bitter fruit just when Parliament's biggest decisions over Brexit are about to be made.
Several Labour MPs have said they are considering their future in the party - but more have said they are sticking with it.
Former shadow environment secretary Mary Creagh told Today: "I have [been approached] and I've said no. I think what is important is we now take a long hard look at ourselves as a political party.
"It is clear that Brexit is pushing both parties to the brink, it is clear that anti-Semitism has taken root in our party."
Edinburgh South MP Ian Murray told the BBC he was sticking with Labour but "may change his mind" unless the party responded to concerns about its culture and direction.
Labour MP Jess Phillips, writing in the Daily Telegraph, called on Mr Corbyn to listen to why the MPs had quit and "act on it", warning that reacting with bitterness could cause the party to "burst apart".
However, shadow foreign secretary Emily Thornberry told the Daily Mirror that the resignations were a "distracting and divisive exercise".
Shadow business secretary Rebecca Long-Bailey told BBC Radio 4's Today programme Labour MPs had to "listen to each other".
But she added: "Equally, I think we also have a duty to unify and make sure that we provide a force for change within Britain."