They are "maximum facilitation" - which is based on using technology to minimise the need for customs checks after Brexit - and "customs partnership" which would involve the United Kingdom collecting tariffs on behalf of the EU.
Today's 90-minute meeting of the Cabinet's Brexit negotiations sub-committee heard presentations from Brexit secretary David Davis and Cabinet Office minister David Lidington on the work completed so far by two ministerial working groups set up by Mrs May last week to look for improvements to the two schemes.
Q&A What is a customs union and why does it matter? But without an agreement on customs, those plans are as yet incomplete. By doing this, they can help avoid the need for costly and time-consuming customs checks during trade between members of the union. A strict customs regime at Dover or between Northern Ireland and the Republic of Ireland would lead to delays that will be costly for business and disruptive for travellers. An Irish peace process built around the principle of entirely unfettered travel between north and south could be jeopardised.
He added: 'Ministers have finally agreed to publish a white paper on the Government's negotiating position, but they still don't know what it will say.
Speaking on BBC Radio 4 this morning Lidington said: "Where we are is when we had an initial talk about this at the cabinet committee, there was some serious criticism made about the technical detail of both the models that were on the table".
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For its part, the EU is putting pressure on Britain to present its preferred options for customs arrangements at a meeting of the European Council in June. The party has used the "humble address" to get the government to hand over documents in the past.
If supported by a vote in parliament, the move would be embarrassing for a leader who has worked hard to keep much of Britain's Brexit negotiations secret, especially the debate on customs arrangements with the European Union. Jeremy Corbyn told his MPs not to expect a change in Labour's stance when the issue returns to the Commons.
"We are going to make our decision on the amendments that come back from the Lords when the government brings the legislation back from the Lords", he said.
Other outstanding issues include guarantees for expatriate rights, agreeing on security cooperation and trade rules after Brexit, to ensure a deal is in place when Britain leaves in March 2019, and an adaptation period ends at the end of 2020.