Highlighting progress in the year-long talks, she will say agreements have now been reached across a broad range of issues including with Spain on the status of the British Overseas Territory of Gibraltar, and with Cyprus on the UK's military bases there. Parliamentarians expressed their lack of confidence in the Conservative leader and support for a referendum on the final terms of any Brexit deal.
A no-confidence vote reportedly looms.
But the withdrawal agreement, or the terms of Britain's divorce, can not be signed off until the two sides agree on the Northern Irish backstop.
Veterinary checks are already conducted on 10 percent of livestock transported to Northern Ireland from the rest of the UK, Barnier said.
The aim of the amendment would be to kill the E.U.'s backstop plan - the insurance plan that seeks to avoid a hard-border between Ireland and Northern Ireland in the event of negotiations failing to deliver a workable solution.
"Contrary to what some are saying, this is not a border we are creating in the Irish Sea to undermine the integrity of the United Kingdom", Barnier told Le Journal du Dimanche.
Last week, the European Union dropped plans for a November summit on Brexit because of a lack of progress in negotiations and began preparations for a disastrous "no-deal" scenario when the clock runs out in March 2019.
The original plan is for a 21-month period starting March 30, 2019, as soon as the United Kingdom has left.
He said: "My preference would be if you had David Davis as an interim leader, not as the leader to take the Tories into the next general election, but to take the party through to Brexit". CNN could not independently verify the number of marchers.
The Plymouth MP slammed May's government for failing to lead over the Windrush scandal, the Grenfell Tower tragedy and Brexit. "The government doesn't have a mandate to gamble with our future", London Mayor Sadiq Khan said at a rally in Trafalgar Square.
Home Secretary Sajid Javid asked the prime minister whether she'd threatened the European Union with no deal, the paper said.
Raising the stakes, the prime minister said the EU's insistence amounted to a threat to the constitution of the UK: "We have been clear that we can not agree to anything that threatens the integrity of our United Kingdom", she added.
Speaking to The House magazine, Conservative MP Johnny Mercer said he would not have stood to represent the party as it is today and described May's government as a "sh*t show".
Former Brexit Secretary David Davis also urged Mrs May to play tough by threatening to ban European airlines from flying over the United Kingdom in the event of a no-deal exit.
"Despite the flawed strategy to date, it is still possible to reset our path towards a proper free trade agreement".
"They know if you wait a period of time and you push hard enough, Theresa May will make compromises and move backwards".
May's most recent attempt to break the deadlock - her suggestion that the United Kingdom could extend the transition period "for a few months" if the United Kingdom needed more time to solve the Irish border conundrum - was met with fury by Tory MPs, who accused May of unnecessarily delaying Brexit while preparing to hand over more money to the EU.
But May faces dissent from her political opponents - and, more worryingly, her own Conservative Party - over her blueprint for separation and future relations with the bloc.
CNN's Erin McLaughlin and James Frater contributed reporting.