The prime minister's plan would see the whole United Kingdom effectively agree to remain in the customs union to help avoid a hard border if no other arrangement can be found.
Mr Rees-Mogg said "There is a concern that the Cabinet is not being fully involved in this, it's getting information at a late stage, there is an appearance of it being bounced, you hear of ministers getting 45 minutes to read crucial documents before Cabinet meetings, this is not a serious constitutional approach".
The European Union's fallback proposal aimed at avoiding a hard border between Ireland and the United Kingdom would effectively keep Northern Ireland aligned with Brussels's customs union and single market.
The EU has made concessions on the biggest hurdle in the talks - the Irish border backstop or an emergency fix to ensure the frontier there stays open regardless of how Brexit goes - but there are still differences on the life span of such a solution, a review mechanism or the depth of regulatory alliance under the all-UK backstop, which has been sought by London.
Brussels has said the United Kingdom will not be able to dissolve the backstop without agreement from the EU.
Foster said: "The Prime Minister's letter raises alarm bells for those who value the integrity of our precious union and for those who want a proper Brexit for the whole of the UK".
Britain's former Brexit Secretary David Davis on Thursday said that members of Parliament will probably vote against Prime Minister Theresa May's Brexit deal.
"The Government will not agree anything that brings about a hard border on the island of Ireland".
The question of how to avoid a "hard border" between Northern Ireland and the Irish Republic after Brexit has proved to be the toughest issue in the exit talks.
Meanwhile, a senior government source said that any reports in the European media that a deal could come in the next few days should be taken "with a very large pinch of salt".
A potential sticking point could be demands for European Union fishing fleets to be given continued access to British coastal waters as the price for agreeing to Mrs May's UK-wide backstop, the Daily Telegraph reported.
"This is a negotiation and needs to be an agreement of course between the British Government, but also with the European Union and the 27 countries that are represented by Michel Barnier and his negotiating team".