The former Prime Minister said the outcome of the meaningful vote could be a deciding factor in Ms May’s future.
But he added that because of the fixed-term parliament arrangement it was “possible” for the Conservative Party to survive in government.
Ministers – who have accepted just one of the 15 amendments to legislation – are confident of winning most of the votes when the EU (Withdrawal) Bill returns to the Commons on Tuesday and Wednesday.
However, pro-EU Tory rebels believe they could be vulnerable on key measures including the customs union and a requirement for Parliament to have a decisive say over what happens next if it rejects a final Brexit deal.
Asked what would happen if the Government lost the customs union vote, Mr Brown told BBC One’s The Andrew Marr Show: “I think what will happen if the Government loses a vote is that there is potentially the chance of a change of prime minister, but I don’t think at that point they will want to give up as a Government, and I think the five-year parliament makes it possible for them to survive.
“So I think the issues come down to whether the meaningful vote, after the customs union vote, the meaningful vote, means that Parliament rejects or potentially could reject the whole settlement, and that’s the point of further crisis.”
Mr Brown said the customs union vote was “difficult for the Brexiteers”, adding that Mrs May’s proposal for a longer transition period “that would mean effectively that we were in the customs union until 2023” would mean Europe becoming the issue of the next general election.
He said: “So I think all these things are still in the air, but certainly there are two moments of decision in the next few months, one on the customs union and one on what’s called the meaningful vote, as to whether when you get the final terms of negotiation, MPs can actually both reject it and perhaps offer an alternative or even discuss another way of consulting the country.”
Asked if it was possible for the UK to remain in a customs union with the EU having left the EU, Mr Brown replied: “I think it is, but I think the issue that we’ve got to resolve is Northern Ireland.
“It is absolutely clear to me that if you want a frictionless border between the north and the south, and everybody says that that is what they want, then you’ve not only got to adhere to some form of customs union or partnership, you’ve also got to adopt some of the single market regulations, therefore this is an issue that I think people have avoided for too long.”
He argued that none of the options being put forward by the Government at the moment were seen as satisfactory to people, adding that their concerns over Europe related to lost jobs, stagnant wages and communities feeling left out.
On Labour’s track record on immigration, he said: “I accept that we did not understand the fears and concerns people would have about the levels of migration in the country.”
He described Labour leader Jeremy Corbyn as a “phenomenon”, but added: “Jeremy himself is going to accept that nobody goes on forever, we’re all phases.”
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