Such a move could trigger a challenge to May's leadership from Brexit campaigners in her Conservative party.
Last week, Britain's upper house of parliament inflicted an embarrassing defeat the government, challenging her refusal to remain in a customs union with the EU after Brexit.
Last week the Lords voted in favour of keeping the option open of a customs union with the EU after Brexit during negotiations with Brussels, and this Thursday a non-binding backbench vote will be held in the Commons on the same issue.
She said it was vital that Britain has "as frictionless border with the European Union as possible" once it leaves the European Union, but added: "We also want to negotiate our own free trade deals around the rest of the world".
Mr Gove later added: "Sajid is right - the referendum vote was clear - we need to take back control of trade - that means leaving the protectionist customs union". "We can make that case to Parliament, we can win that case".
Questions on the Customs Union arise as the EU rejects the UK's proposals for the Irish border after Brexit.
Britain's Prime Minister Theresa May speaks at a news conference to mark the end of the Commonwealth Heads of Government Meeting at Marlborough House in London, Britain, April 20, 2018.
The scheme would see the United Kingdom act as the external frontier for the European Union, with the United Kingdom planning for customs authorities to charge whichever is higher of British or European Union tariffs on good entering the country.
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A source familiar with the discussion told the paper: "The discussion focused on what to do if parliament votes to stay in a customs union".
Under the rules of the customs union, the European Commission in Brussels strikes deals for the whole bloc.
Chancellor Philip Hammond and Business Secretary Greg Clark are said to favour remaining in the customs union.
What is the argument for staying in the customs union?
In a letter to the Prime Minister, seen by HuffPost UK, Shadow Brexit Secretary Sir Keir Starmer has offered to give up parliamentary time set aside for Labour to ensure a meaningful vote on the issue.
CBI business groupWhat is the argument for leaving the customs union?
Reports over the weekend suggested a "wargaming" exercise into the consequences of a concession showed that not even leading Brexiters such as Michael Gove, the environment secretary, or Boris Johnson, the foreign secretary, would resign.
He told the BBC: "Most members of the public woudld be forgiven for saying they'd never heard about a customs union".
Though it may anger some of the more fervent Brexiteers in the party, there was support for May from Stephen Kerr, one of the Scottish Tory MPs.