He declined to go into any specifics about the kind of legally binding changes Cox was trying to secure.
In January, MPs rejected the withdrawal deal she has reached with the European Union by 230 votes - the biggest defeat for a sitting government in history.
Parliamentarians were concerned that the backstop - which would see the European Union effectively annex the British province of Northern Ireland for customs and regulatory purposes while mainland Great Britain would be incorporated into an EU-controlled "singles customs territory" - is not time-limited, and the British government would not be allowed to leave it without the EU's permission.
But the Labour leadership see this as a "bribe" to tempt some of their own MPs to break ranks and back Mrs May's deal. EURACTIV has travelled to Northern Ireland where Brexit is more than a political or economic issue - it is a challenge for peace.
Soubry said: "Voters will not be fooled - especially those in areas which voted Leave and are now demanding a people's vote because they know, whichever way you do it, Brexit will harm their futures".
The EU's chief Brexit negotiator Michel Barnier will meet Britain's negotiating team tomorrow, an EU spokesman said, as both sides seek a breakthrough in the Brexit backstop.
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In a list of terms sent to May - which Fox called a "genuine attempt to find common territory" - members of the hardline ERG called for a "clearly worded, legally binding treaty clause" that overrides the text of the withdrawal agreement; language that moved beyond restating the "temporary nature of the backstop" to Cox's legal advice that it would "endure indefinitely", and a "clear and unconditional route of out the backstop if trade talks fail", including through a time limit or unilateral exit mechanism.
"Labour pledged in 2017 that we will establish a network of regional development banks that will be dedicated to delivering the finance that our small businesses co-operatives and innovative projects need across the whole country. We are calling all trade unions, all representative groups, civic leaders and citizens to join with us and demonstrate our anger at being taken out of the European Union".
The EU is said to be refusing to countenance Cox's suggestion that backstop arbitration should be overseen by an "independent" body outside the jurisdiction of its own Court of Justice, however, making it hard to see how any tweaks to the mechanism now laid out could be anything but cosmetic.
Shadow chancellor John McDonnell has signalled Labour will whip its MPs to back a bid for a new Brexit referendum.
"The Attorney-General continues to pursue legally binding changes to the backstop that are necessary to ensure it can not be indefinite", a Downing Street spokesman insisted.
"I urge my Labour colleagues to consider voting for an improved deal", she told Sky News.