Britain is due to leave the bloc in less than three weeks, on March 29.
"If this deal is not passed, then Brexit could be lost", May said.
At a joint news conference, May and Juncker claimed to have succeeded.
"We really want to be over with it now".
THE British government says it has secured "legally binding changes" from the European Union to overcome a key stumbling block on the Brexit deal.
Merkel deflected a question on whether she was prepared to delay the Brexit deadline. 'However, the matter of law affecting withdrawal can only inform what is essentially a political decision that each of us must make'.
Their group of lawyers will meet to analyse whether they think Britain can exit the backstop or not.
On Monday, the government admitted that talks were "deadlocked" but insisted that it is set to go ahead, despite pressure from some quarters, for the Tuesday vote to be downsized.
A so-called Star Chamber convened by the Leave-backing European Research Group found that agreements reached by the Prime Minister in 11th-hour talks in Strasbourg do not deliver the legally-binding changes the Commons has demanded.
Notably, betting markets have raise the odds of tonight's vote which favours a "no deal" Brexit passing through parliament from 10% to 30%.
"What we are hearing is the biggest change in terms of trade this country has faced since the mid-19th century being imposed on this country with no consultation with business, no time to prepare", she added.
"We are hoping we will be able to win that vote but that does depend on the European Union coming to the table and taking seriously the (UK's) proposals".
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Brexit-supporting lawmakers expressed suspicion at the haste of May's last-minute assurances and suggested a delay to allow sufficient analysis of them and Cox's advice.
Placing tariffs on imports would likely raise the cost of goods, which would put pressure on United Kingdom consumers in the event of a "no deal".
"The uncertainty around Brexit is already worrying citizens".
Minutes after Cox's advice, the pound was 1.1 percent lower at $1.3014, nearly two cents down from where it was earlier.
Juncker said May and he have agreed a "legal instrument" to ease British concerns over the backstop. The government said this was to "ensure access" to those markets is "maintained".
Britain s House of Commons overwhelmingly defeated the deal in January and was expected to do so again on Tuesday without meaningful change.
UK's chief legal advisor Attorney General Geoffrey Cox said the extra assurances won by May do "reduce the risk that the United Kingdom could be indefinitely and involuntarily detained" in the backstop if talks on the two sides' future relationship broke down due to "bad faith" by the EU.
"Politicians have proven themselves to be incapable of resolving this impasse in parliament".
Senior British government ministers have warned rebellious lawmakers that if May's deal is voted down then there is a chance that Brexit could be thwarted.
The three-part package of changes effectively aims to resolve a key sticking point for British MPs over the so-called backstop plan to keep open the border between European Union member Ireland and the British province of Northern Ireland.
The agreement's resounding defeat drew a smattering of applause.
Due to the sensitive nature of the region and because the free passage is one of the crucial articles of the 1998 Belfast Agreement, the U.K., EU, Northern Ireland and Ireland all reject the idea of returning to a hard border where checkpoints and customs buildings will need to be installed.
"If she doesn't she will forfeit the confidence of the House of Commons".