The extension would give United Kingdom more time to come up with a workable customs solution.
May has been struggling to unite her cabinet over the terms of Britain's divorce with the European Union, with a row over future customs arrangement dividing her government and all but stalling Brexit negotiations.
The agreement could see Britain remain within the EU's common external tariff until a future customs arrangement is ready in a temporary fix created to avoid a massive jolt for business and a hard border in Northern Ireland.
May said three of her objectives - to forge an independent trade policy; a frictionless trade with the European Union and avoid disruption between Ireland and Northern Ireland - were not easy to achieve but maintained that only her government will fulfil the expectations of the pro-leave voters of the 2016 referendum.
The news follows weeks of Cabinet deadlock over the UK's future customs arrangements.
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The premier has tasked Bradley and Business Secretary Greg Clark to work to improve the max fac plan in conjunction with Brexit Secretary David Davis, the only minister of the three to support it. Environment Secretary Michael Gove and Trade Secretary Liam Fox are focusing on the partnership plan with May's de facto deputy, Cabinet Office Minister David Lidington, himself the only supporter of the option of the three.
HuffPost UK understands that Brussels sees serious problems with the plan, including the risk that exemptions would lead to a growth in small firms that could avoid strict European Union rules on standards on everything from toy safety to pesticide strength.
Bradley said that the advantage of the partnership plan would be not changing the way the common external tariff on goods is collected, enabling easier cross-border trade between Ireland and Norther Ireland.
To soothe concerns of Brexiteers, it would include a "sunset" clause ensuring Britain does leave in the end. Both sides fear a return of border controls could reignite the violence that afflicted Northern Ireland until a peace deal in the late 1990s.
Max-fac would take several years to set up but the customs union extension would allow extra time for its implementation.
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She also singled out the British government's "practical" stance pertaining to the ways to grapple with the sensitive issue of Northern Irish border.
Another spokesman for May added: "We have been clear we are leaving the customs union, a customs union".
A pro-EU Cabinet source said the pair had been "outgunned".
May told reporters, answering questions about if Britain would leave custom union with Brexit, "No". The horizon seems to be unreachable.
"Are you saying we might not consider putting up or using cameras away from the border ... because of some blackmailing and threats by dissidents who might actually decide that they are going to start killing people?"
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