EU supporters, meanwhile, are pinning their hopes on a European Court of Justice ruling on Monday on whether Britain s parliament has the right to unilaterally halt Brexit in its tracks.
An updated House of Commons schedule said there would also be a statement on "business of the House" after May's address, indicating a sudden change to the parliamentary timetable.
The EU's executive said on Monday it would not renegotiate its Brexit deal with Mrs May.
The same newspaper reported that another parliamentary private secretary was on the verge of quitting, and said Cabinet Brexiteer Penny Mordaunt was considering whether to back the deal or resign.
May indicated she was seeking further assurances from the European Union on the working of the backstop - an insurance policy to avoid a return to border checks between the British province of Northern Ireland and the EU-member Irish Republic, and would seek to give the British parliament more power over its application.
As May made her statement, the British pound dropped to its weakest level since June 2017, falling to $1.2622.
May too was not comfortable with all elements of the deal, including the "backstop" arrangement to guarantee no return to a hard border between Ireland and Northern Ireland, Hunt said.
"The deal we achieved is the best possible".
The ECJ could also decide Brexit can only be postponed or suspended by unanimous agreement of the other 27 nations.
Failure to approve some version of May's deal in coming weeks would put Britain closer to dropping out of the European Union in March with no deal, a worst case scenario that most economic commentators, including the Bank of England, say would lead to a savage recession.
A delay will force the weakened government to try to renegotiate its departure or face two equally unpalatable choices: a "no-deal" Brexit that could bring havoc to the economy or call a second referendum, which could undermine faith in the country's democratic institutions and lead to mass protests. He added that May is confident that she will take all the votes that are needed to pass the Brexit deal.
Coveney, who was in Brussels to attend a meeting of foreign ministers alongside Hunt, said Ireland is "watching closely" developments in the United Kingdom parliament on the Brexit deal.
"We are going to be glued to our trading screens or television wherever we are", said Neil Jones, head of hedge fund foreign exchange sales at Mizuho, another Japanese bank, in London.
"The government has lost control of events and is in complete disarray", said opposition leader Jeremy Corbyn. May has shot down suggestions of a second referendum on the Brexit deal.
"We voted very clearly".
"We view the situation with a mixture of worry and hope", one FTSE CEO said.
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