He said it was "critical" to have a new customs arrangement ready for the end of the transition period in January 2021. The Sunday Telegraph also reported that at least 12 members in the full cabinet of 28 are set to oppose it.
Health Secretary Jeremy Hunt appeared to endorse the idea yesterday.
The Tory MP, who is also chair of the Treasury Select Committee, also criticised the other proposal being put forward, and said: "It seems to me that's what's called the maximum facilitation - which seems to rely on future technology not yet invented - would absolutely basically be a deal in name only because it doesn't talk about an enduring relationship with the European Union, which I think is what the Prime Minister said she wanted to create, and it causes enormous problems on the island of Ireland". 'This is a very, very hard situation that we're in as a country but it's got great opportunities for us.
"As in any negotiation, there will have to be compromises", May wrote.
"I will need your help and support to get there".
"I think the crunch really is about the Northern Ireland border because unless the Government is prepared sooner or later to face down the Irish Government and the European Union threatening infrastructure at the Irish border and call their bluff on that one, then we really haven't got much of a negotiating position". It would also mean close alignment of regulations.
The MPs, who support Brexit, demanded May drop a proposal for a customs partnership with the European Union once it leaves the bloc, arguing that the partnership would make meaningful trade deals "impossible".More news: Facebook Suspends 200 Apps Over Mishandled User Data
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But Mr Gove said: 'Across Government, across Cabinet, there is agreement that neither of these two models is absolutely ideal. "Because it is novel, because no model like this exists, there have to be significant question marks over the deliverability of it on time", Gove said. "It is my view that the new customs partnership has flaws and that they need to be tested", he added.
He rejected the possibility of delaying the customs deal, a proposal raised by Tory MP Nick Boles last week.
He insisted the Government needed to "crack on" with finding a solution as "in delay there lies no plenty, as Shakespeare once said".
He said Ms Thornberry had a chance to "part those clouds of confusion", to which Ms Thornberry replied: "We're quite willing to exchange places with the other side, all we would ask for is they call a general election".
May last week ordered her ministers to take responsibility for resolving the issue themselves, splitting her inner cabinet into two working groups to iron out their differences.
They are expected to make a presentation to the Brexit "war committee" tomorrow and submit a written paper next week - meaning no decision is likely this week.
Simon Coveney, the Irish foreign minister, dismissed the idea of using any form of infrastructure or technology to maintain separate customs regimes between the Republic of Ireland, an European Union member, and Northern Ireland, which will become a non-EU member after Brexit. May has insisted that the final Brexit deal must honour the agreements in the Northern Ireland peace process and not create any hard border between Northern Ireland and the rest of the United Kingdom, creating the prospect of impasse over the issue.