In a note of criticism of Mrs May's strategy, she said there appeared to be a lack of "ownership" in Britain of the agreement struck between the two sides in November, and that there was insufficient transparency in the prime minister's moves.
Their plans could set a new direction for the country, or they could underline the debilitating lack of consensus that persists on the most divisive issue the country has faced for generations.
One of them proposes scrapping the backstop.
The dispute between Madrid and London has already threatened to derail the whole Brexit negotiation process, but a last-minute deal with Prime Minister Pedro Sanchez cleared the way in November for the 27 European Union states to approve the divorce package.
May told Parliament after the vote that "it is now clear that there is a route that can secure a substantial and sustainable majority in the House for leaving the European Union with a deal".
Three proposals have gained a head of steam among MPs.
DUP leader Arlene Foster - whose 10 MPs prop up the minority Conservative administration in the Commons - said it provided a "feasible" alternative to the backstop.
Sinn Féin has said there is no room to change any aspect of the backstop in the Brexit Withdrawal Agreement and that Irish politicians would not be to blame in the event of a hard border in Ireland resulting from a "no deal" Brexit.
European Commission vice-president Jyrki Kateinen said there was "no reason to give any concessions" to the United Kingdom and there was "not much room for manoeuvre" on the backstop.
As MPs prepared for a crucial series of Commons votes which could shape the next phase of talks with the EU, International Trade Secretary Liam Fox indicated Mrs May was ready to reopen the Agreement in order to secure a legally-binding text.
The developments came as major supermarkets and fast-food chains warned that a no-deal Brexit would force up prices and reduce the choice and quality of produce for shoppers.
What influence do the amendments have?
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A packed week for markets got underway with investors bracing for large-cap tech earnings, U.S. The trade meeting is just one of several big events that could swing markets in a busy week.
British stocks jumped Tuesday and the pound was firmer ahead of the votes. However, Britain could not leave the backstop without the EU's consent.
Weber said: "If there is now a unilateral attempt to reopen the agreement, the effect will be that not just the backstop has to be renegotiated - then the Gibraltar question, the question of how much money Britain has to pay for exiting, the question of citizens' rights will have to be renegotiated".
Labour MP Chuka Umunna, who has pushed for a second referendum, told Sky News that the torpedoing of the Cooper amendment, and another similar amendment, represented "a bad day for Parliament".
At the weekend, Ireland's foreign minister, Simon Coveney, stated baldly the backstop simply "isn't going to change".
The retailers say that they are already stockpiling where possible.
"We stand by Ireland", he said.
Today's votes will not mark the end of Britain's turmoil over Brexit.
"This is not a Brussels day, this is a London day", said European Commission spokesman Margaritis Schinas. Speaker John Bercow will likely announce the amendments selected for voting around 12:45-13:00GMT with MPs set to vote on the amendments from 1900GMT.
There is no guarantee any of the frontrunner amendments will pass.
Markets have bid Sterling up over recent days on the view that Cooper's amendment is an insurance policy against a "no deal" Brexit taking place, based on the assumption there is a parliamentary majority against a "no deal" Brexit. If she failed to do so, the parliament would then vote on asking the European Union to extend Brexit deadline from March 29 to December 31.
Aside from Brady's proposal, the other key amendment expected to garner a large amount of support, possibly enough to be approved by parliament, is one put forward by opposition Labour lawmaker Yvette Cooper.