The UK government is so far sticking to plans to leave both the single market and customs union.
The defeats also came not long after downing street were playing down claims over a split within the government over the customs union.
An opposition-backed amendment to the legislation, on which the government suffered two defeats last week, was backed by 316 to 245, a majority of 71.
Johnson has already said that staying in a custom union would be more awful than staying in the EU.
Speaking to the MPs, Davis said confirmed it any UK-EU deal would collapse if parliament voted against it. It suggests that the United Kingdom accepts it may have to agree to an unpalatable backstop plan for keeping Northern Ireland - and possibly the whole United Kingdom - in many parts of the customs union and single market rules indefinitely.
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'In particular, the four freedoms of the single market go together.
Harvey Blackwood analysts say the United Kingdom is attempting to secure a detailed free trade agreement with the European Union and is aiming for it to be signed not long after it officially leaves the bloc less than a year from now.
"If we leave the European Union, we leave the customs union and we don't have anything to replace it with, we suddenly, when we leave, are going to have to make not just new deals with countries across the world for free trade but also all the other ones that Europe had made until that point".
Brexit Secretary David Davis suggested the thorny question of how to avoid a hard border with Ireland might still not be resolved when the United Kingdom leaves the European Union next March.
The defeat on Monday is one of several Theresa May's government is expected to face in the Lords as parliament debates legislation that will enact Britain's exit from the European Union in March next year, but the votes, while embarrassing for the Tories, can be overturned by the House of Commons.
May's difficulties in resolving the Northern Ireland border issue. For example, ten pro-European Tories have laid down an amendment to a Brexit...
Mr Davis tweeted: "As we leave the European Union it's essential both the United Kingdom and European Union do what it takes to keep the border, which I saw this morning, free from physical infrastructure".