However, Mrs May's DUP allies poured cold water on the suggestion and the Prime Minister stressed she would not be prepared to accept anything that resulted in Northern Ireland effectively being "carved away" from the rest of the UK.
"But there is still a lot of work to do on the question of how future trade relations will look", she added.
May's proposal calls for Britain to remain in a free trade zone with the European Union for agricultural and manufactured goods, that would be covered by a "common rulebook" of regulations after Brexit.
May told reporters that her offer to solve the border riddle - the biggest obstacle to a deal - was "the only credible and negotiable plan on the table that delivers no hard border in Northern Ireland and also delivers on the vote of the British people" to leave the EU.
As reported by RTÉ News yesterday, the EU's chief negotiator Michel Barnier will suggest that of the four categories of checks and controls that would normally apply between the European Union and a third country, only live animals and agri-food checks would need to take place at British ports on goods going to Northern Ireland.
With Britain's departure from the European Union on March 29, 2019 looming, there are growing concerns that a deal on the post-Brexit relationship may not be cobbled together in time to ensure a smooth and orderly British exit.
Writing in Germany's Die Welt, May said the two sides were "near to achieving the orderly withdrawal that is an essential basis for building a close future partnership".
"We have to do everything to avoid a hard Brexit", he said.
At a summit dinner at the Felsenreitschule theater - known to film fans for a scene in the musical "The Sound of Music" - May will make another pitch for support for her Chequers plan.
On Wednesday, Tusk announced a special Brexit summit in November, when he hopes a deal will be "finalized". Tusk has said that October will be the "moment of truth".
Austrian Chancellor Sebastian Kurz, the host of the Salzburg summit, backed May in saying that "both sides need to compromise". "I am still optimistic".
Mrs May remains wedded to the Chequers blueprint, noting it is the only proposal on the table as the deadline approaches, although she indicated the United Kingdom will unveil new measures on the future status of the Northern Irish border in a bid to break the deadlock.
"On Brexit, it's very hard. There is no progress", Lithuanian President Dalia Grybauskaite told Reuters.
The EU has proposed that Northern Ireland, a British province, continue to follow many EU trade rules and regulations to maintain the status quo with Ireland, a remaining bloc member.
As EU leaders prepared to gather in Salzburg, Michel Barnier said he was working on an "improved" plan to avoid the return of a hard border while respecting the territorial integrity of the United Kingdom in an attempt to break the talks deadlock.
"The Chequers plan can not be take it or leave it". They will discuss the issue among themselves over lunch on Thursday, setting what Barnier hopes can be a path to a final deal in two months.
Asked if the EU's criticism of her plans increased the prospect that the two sides would fail to reach a deal, May said: "We are continuing to work for a good deal".
However, at the gathering in Austria, Mrs May insisted there would be no delay to the UK's March 2019 departure, no second referendum and therefore the onus was on the continent's leaders to find a solution if they wanted to avoid a no-deal Brexit.
"So there is a danger of that happening if Chequers doesn't prevail".