It was claimed that the European Union had last week agreed a major concession to unlock a Brexit deal, by accepting that checks on goods could take place in factories and shops rather than at the border.
Downing Street has called the reports speculation, but also claim that the majority of the UK's Brexit plan had already been agreed.
But Britain's Conservative government and its Northern Irish ally, the Democratic Unionist Party, won't accept that because it would mean customs and regulatory checks between Northern Ireland and the rest of the United Kingdom. That in turn could sway the euroskeptic wing of her Conservative Party.
Under the banner headline "May's Secret Brexit Deal", the newspaper said she's also on course to gain an agreement on a "future economic partnership" that will allow Britain to keep open the prospect of a similar free-trade accord to the one Canada has with the EU.
Brexit is undermining Northern Ireland's hard-won peace by creating tensions between Catholic and Protestant communities, Irish Prime Minister Leo Varadkar said Saturday, even as hopes rose for a solution to the Irish border problem that has deadlocked negotiations.More news: Raheem Sterling not scared anymore, says Manchester City boss Pep Guardiola
Mr Brokenshire, appearing on television, was asked if a deal was close, replying: "Well, we want to get that deal, we're obviously working hard to see that that happens".
Her plan would avoid the need for a hard border in Northern Ireland, which has always been the main sticking point in negotiations.
"We hope a deal can be done but we're not there yet".
But the government's Brexit department stated that they are confident there will be a deal that works for businesses - and reiterated their stance against a People's Vote.
It came as more than 70 business leaders backed a referendum on the terms of the Brexit deal, warning that the United Kingdom faces "either a blindfold or a destructive hard Brexit" that would be bad for both firms and jobs.
More than 70 business leaders - from lastminute.com founder Martha Lane Fox and former J. Sainsbury PLC chief Justin King to Cobra Beer founder Karan Bilimoria and ex-chairman of Rolls-Royce Holdings PLC Simon Robertson - signed a letter arguing that both the government's current plans for Brexit, and a no-deal Brexit, would be bad for companies and jobs.