But that will be too little, too late to conclude a deal this week, according to ministers attending the EU's foreign affairs council in Luxembourg on Monday.
"I fully appreciate the risks of a "no deal" but the dangers of a bad deal are worse", she wrote.
Hunt hosted eight eastern European counterparts at the foreign secretary's official country residence in Chevening, southeast of London, on Sunday.
Only two days ahead of a summit once seen as the moment when Britain and the European Union would have to reach a Brexit deal, both sides are still refusing to blink over the question of the Irish border.
It opposes any new checks on goods between Northern Ireland and Great Britain and has threatened to pull support from the government over the backstop issue.
To nods from Boris Johnson and other Brexiter MPs, Greening said May's plan "represents a unique loss of sovereignty for Britain and that for the first time we will have tied the hands of future generations to be bound by rules they will have had no chance to write".
"I am clear we are not going to be trapped permanently in a single customs territory, unable to do meaningful trade deals".
Following three days of talks with key figures in Brussels, DUP leader Arlene Foster - whose party props up the Government at Westminster - issued a renewed warning that they could not accept the European Union proposals as they stood.
Preventing any return of a "hard" border in Ireland has become one of the major obstacles, with Brexit campaigners fearful that a backstop with no expiry date will keep Britain inside a customs union with the EU indefinitely.
The invitation came just hours after May came under attack from all sides in the Commons as she defended her Brexit negotiating plan.
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In an effort to find a way to avoid a hard border between Ireland and Northern Ireland after Brexit reports have suggested the United Kingdom could remain tied to Brussels' rules beyond the end of 2020 to give negotiators more time to finalise a trade deal.
Insisting that a Brexit divorce deal was "achievable", May said the border dispute should not "derail the prospects of a good deal and leave us with the no-deal outcome that no one wants".
Sammy Wilson, the DUP's Brexit spokesperson, said it now believes a no-deal Brexit was nearly inevitable, and described the talks in Brussels as turning into a "battle for the union".
"It is time to scrap the backstop, and simply agree what is manifestly the case - that no one wants any new physical checks at the northern Irish border, and nor is there any need for them".
The Prime Minister will have a cabinet meeting on Tuesday before she heads to Brussels and she may struggle to get a consensus.
Britain has proposed sticking with European Union customs rules after Brexit as a fallback option to keep the border open, until a wider trade deal is agreed that avoids the need for frontier checks.
'We are entering the final stages of these negotiations, ' she said. But they were disappointed by the issue that has dogged the talks for months - how to ensure that no hard border is created between the EU's Ireland and Britain's Northern Ireland once Brexit happens on March 29.
Talks are stuck on the issue of the Irish border and a guarantee, or so-called backstop clause in the divorce treaty, which is proving impossible for May's allies at home to accept. "Whether or not we do this week, who knows?"
He told reporters "a backstop can't be time-limited", adding: "The backstop will be there unless and until something else is agreed, but unless you have something to replace it well then the backstop needs to be there as an insurance mechanism".