May wrote a conciliatory letter to opposition Labour Leader Jeremy Corbyn, after he proposed setting her a new deadline of Feb 26 for winning lawmakers' backing for a final exit deal.
Discussions about a wide-ranging trade deal with Japan have also hit a hurdle after Japanese negotiators said they wanted to start from scratch, rather than mirroring the terms of the existing European deal.
Britain is due to leave the European Union on March 29, but Parliament has rejected May's divorce bill, leaving the prime minister to seek changes from the EU.
"We can't allow that to happen", Sir Keir said.
Speaking afterwards, May said that the UK "must secure legally-binding changes to the Withdrawal Agreement to deal with the concerns that Parliament has over the backstop", adding, "Taking that together with the other work that we are doing on workers' rights and other issues will deliver a stable majority in the UK Parliament".
"If the meaningful vote has not happened, so in other words things have not concluded, then Parliament would have that further opportunity by no later than 27 February", said Mr Brokenshire.
Trade with Switzerland is worth around £32 billion a year and the rollover of the arrangements come as Dr Fox is under pressure to secure continuity for other existing European Union deals.
The government maintains that these agreements will only require renegotiation if the United Kingdom leaves without a deal - and that otherwise they will automatically be replicated.
Labour's Brexit policy chief Keir Starmer told the Sunday Times newspaper that his party would seek to use the debate in parliament this week to prevent May from waiting until the last minute to come back with a deal, and compel her to present a fresh accord for lawmakers to consider before February 26.
May has been seeking changes to the deal with Brussels since it was rejected by a record majority in parliament on January 15.
Blair warned Sunday that a no-deal Brexit would inevitably lead to a so-called hard border being reintroduced on the island of Ireland with "potentially devastating" consequences for the peace process in Northern Ireland.
British government asked lawmakers on Sunday to give Prime Minister Theresa May more time.
A number of government ministers will also be meeting their counterparts across the continent this week, in order to underline Mrs May's determination to achieve a deal. The UK government has dropped its initial demand that they retain easy access to the single market.
"It will be the decisions that businesses take about jobs and investment - and they will reduce them - so you have less potential to trade globally, that will mean less investment in the future and that will mean fewer jobs in the future", she told Sky News.
Cardi B Makes History at Grammys
But nothing could compare to the raw emotion she showed in her acceptance speech. "Trash", she continued, and "literal bulls-". She lost out on her other four nominations, including in the record and album of the year categories.
In as much as most movies show how gambling is bad or addictive, look on the brighter side, it is a good way of making money. With so many people shaming gambling, directors are trying to put it out there that it is not as bad as people make it seem.
Samsung shares hurt by negative Morgan Stanley forecast
The company will be starting construction soon on a special EUV facility as part of their Hwaseong complex in South Korea. The Note 8 has seen an incredible return to form for the company and saw over 4 million devices sold around the world.
What Are Online Games
If we are all honest with each other, there is no denying that when real money is involved the activity is no longer for leisure. And in some cases, it is even used to refer to something that is done elsewhere for an income but now being done for recreation.