The amendment was rejected by MPs 85-334, a majority of 249.
The party's 10 MPs have rejected the backstop arrangement now in place because they feel it risks giving Northern Ireland a separate status to the rest of the UK.
The draft paper reportedly outlines that if the United Kingdom seeks an extension to Article 50 of more than three months than it must take part in the election.
The motion to extend the UK's exit was put forward to MPs at Westminster yesterday (Thursday) and was passed by 413 votes to 202. Speaking to the BBC's Political Thinking podcast, Ms McVey said: "The element now is that people will have to take a bad deal rather than no deal".
If she is to win, one leading indicator could be whether the 10 DUP MPs, who prop up her minority government, also get behind her deal.
Her deal, an attempt to keep close relations with the European Union while leaving the bloc's formal structures, was defeated by 230 votes in parliament on January 15 and by 149 votes on March 12.
Boulogne-Billancourt : Elle découvre une souris dans le pain qu'elle achète
Des contrôles d'hygiène et des contrôles sanitaires sont réalisés de manière très fréquente, en particulier sur ce magasin. Une cliente a découvert une souris vivante dans un emballage de baguette de pain en France.
EU Council President Donald Tusk, who coordinates the strategy of European leaders, said in a Twitter post Thursday that he wants government chiefs to be open to a "long extension" if Britain decides to "rethink its Brexit strategy and build consensus around it".
There is some controversy about the third "meaningful vote", given that, ordinarily, parliament would not be able to vote on the same thing a few times. They didn't believe and neither would I have believed - having got warm words from people in authority "we will do this", and remember the 100-plus times "we will be out on 29 March" - absolutely not.
"If it is not the case what we need is clear vision from the UK Government how much time the UK needs to come up with new proposals, new ideas how we proceed".
Tellers reveal the result of the second meaningful vote.
Further talks are expected over the coming days with the DUP in the hope that persuading the ten Northern Irish MPs to back the deal will help sway scores of Tory Eurosceptics to fall into line.
The draft paper obtained by the Financial Times makes clear Britain has to take part in the May 23-26 votes if it wants an extension of more than three months.
There is no path to victory for May without winning many more of the Brexiteers' votes. It may be that the government will allow MPs to have a further debate on the sort of Brexit they'd like to see, or that MPs will continue to amend a government motion to propose alternative Brexit options.