They reportedly agreed that Britain should stay aligned to the customs union if highly complex and untested technology needed to operate borders after Brexit is not ready, but made clear that Britain would still be able to strike its own free-trade deals.
"We have set three very simple objectives for a future customs union", May said at the weekly session of prime minister's questions in the House of Commons.
The development came after May stressed that the United Kingdom government would ensure the absence of a hard border between Northern Ireland and the Republic of Ireland after Britain's withdrawal from the EU.
British cabinet ministers are deadlocked over a future deal with the block and the Telegraph newspaper said Britain would tell Brussels it was prepared to stay in the customs union beyond 2021, sending the pound to a two-day high.
European Union leaders meeting May in Sofia on Thursday were "in listening mode" and hoping for reassurances from her, said one official, before a formal summit in June when the sides want to mark another milestone in the negotiations.
Britain is due to leave the European Union in March next year although it has secured a transitional arrangement to keep its trade ties with the bloc unchanged until the end of 2020, as long as a permanent deal can also be reached in the coming months.
Britain otherwise risks crashing out of the bloc, a scenario that could hurt the economy and disrupt people's lives.
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"It is an absolute redline for us that there could not be a hard border on Ireland", Irish Prime Minister Leo Varadkar told reporters in Sofia.
But the proposals have not gone down well with hard Brexiteers who want a "maximum facilitation" customs option. Both sides fear a return of border controls could reignite the violence that afflicted Northern Ireland until a peace deal in the late 1990s.
"We need that to be part of the withdrawal agreement, and if its not then there will be no withdrawal agreement and no transition period".
Mr Varadkar said he was "really looking forward" to meeting with Mrs May.
She also singled out the British government's "practical" stance pertaining to the ways to grapple with the sensitive issue of Northern Irish border.
Cabinet office minister David Lidington admitted this morning that the government had asked lawyers to look at the two post-Brexit customs options to check if they are legal. "They are undermining the Good Friday Agreement and denying democratic rights to Irish citizens without any justification whatsoever".
At the end of last month, EU's chief Brexit negotiator Michel Barnier said there could be no Brexit withdrawal agreement without the backstop option for the Irish border.