The votes were not legally binding for Prime Minister Theresa May's government, which has seen its attempts to push through the PM's Withdrawal Agreement defeated on three separate occasions since January.
Donald Tusk, President of the European Council, appealed for the EU to give Mrs May time as she attempted to break the deadlock.
"As things stand now, the no-deal option looks likely".
Hinds tells the BBC that the deal "is a good balance, and I hope colleagues can get behind it".
Her decision to call an unusually long cabinet meeting Tuesday further fueled expectations that Britain might be heading for its second election in as many years. A European diplomat said: "A long extension will be conditional on loyal membership obligations".
The negotiations with Jeremy Corbyn have provoked a fierce backlash from many in the Conservative Party, however, with two junior ministers resigning from the government. It is the first time she has committed to following the instruction of lawmakers.
He said Britain could still accept deal agreed by May, reiterating it was "the only way" for Britain to leave the bloc in an orderly way.
"I have failed, chiefly because my party fails to compromise", Boles said.
He told May that Labour wanted a customs union with the EU, access to its single market and raised "the option of a public vote to prevent crashing out or leaving on a bad deal".
May's move infuriated pro-Brexit politicians, who say Britain must cut ties to the European Union in order to forge an independent economic policy.
Rank-and-file Tory Bill Cash called the bill an "utter rubbish dump", while his Conservative colleague Mark Francois complained it had been "rammed" through the Commons "in barely four hours". She added that if they couldn't reach an agreement, the government would put a range of options to MPs.
Businesses have warned that the economic impact in Britain could be devastating.
Prime Minister Theresa May has previously ruled out two of the ideas - remaining in a customs union with the EU, or holding a second referendum.
But it has been roundly rejected by lawmakers on both sides of the Brexit divide.
Labour MP Yvette Cooper and her allies are seeking to force Prime Minister Theresa May to request the European Union give a further extension of Article 50, the section of the European Union treaty under which Britain asked for an exit.
In a letter to the prime minister he wrote: "I simply can not support any further extension to Article 50".
Varadkar stressed "there's still time" for May to come to the April 10 summit with "credible" proposals.
"If we move quickly this week and we get this deal over the line it is still possible that we may be able to avoid having to have those European Parliament elections (in May)", Hinds said.
Cooper has introduced legislation, which Parliament is set to consider, this week, that would require May to seek to extend the Brexit process beyond April 12 in order to prevent a no-deal departure.
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