Meanwhile, Michel Barnier has said the EU proposal "limits itself to what is absolutely necessary to avoid a hard border"
The DUP has warned Theresa May it will pull support for her government if it doesn't approve of any Brexit deal reached with Brussels.
Currently, the Northern Irish party is propping up Mrs May's government in Westminster - with the party's nine MPs in a confidence and supply arrangement with the Conservatives.
The DUP has frequently insisted it will not support any deal that treats Northern Ireland differently to the rest of the UK - despite the insistence from the EU that a previously agreed 'backstop' arrangement is necessary to avoid a hard border in Ireland.
Ahead of a key meeting of EU leaders next week, media reports have suggested that progress is being made in the contentious Brexit negotiations - including over the backstop.
However, senior DUP leaders have warned they could pull support for the current government if they don't approve of a proposed deal.
If they vote against the upcoming budget, it could potentially collapse Theresa May's government.
Conservative MP Steve Baker - a Brexiteer - suggested the DUP is 'absolutely furious' over the reported proposals.
DUP absolutely furious tonight— Steve Baker MP (@SteveBakerHW) October 10, 2018
Describing the backstop proposal as “shameful”
And NI’s probable exclusion from trade deals as “disgraceful”
No one should underestimate their resolve
In an article in the Telegraph, the DUP Brexit spokesperson Sammy Wilson insisted his party "will not be bullied into propping up a soft-touch Government which gives in to the EU's demands".
He argues: "The road which the leaks and briefings is outlining is the road to parliamentary defeat for any deal which the Prime Minister brings forward."
He adds: "[Mrs May] will not have DUP support regardless of whether the Government tries to bribe, bully or browbeat us into accepting it.
"However, if the Government decides to stand up to the outrageous, capricious and extortionate demands from Brussels, she will have our full support and she could rally the country behind her."
Amid the warnings, it's reported that Theresa May is due to meet with senior cabinet ministers today to update them on the progress in Brexit negotiations.
The EU's chief Brexit negotiator Michel Barnier, meanwhile, has outlined some of the backstop proposals.
In a speech yesterday, he explained that the "only visible systematic checks" between Northern Ireland and the rest of the UK would involve scanning bar codes of lorries or containers on ferries or at ports.
He observed: "There will be administrative procedures that do not exist today for goods travelling to Northern Ireland from the rest of the UK.
"Our challenge is to make sure those procedures are as easy as possible and not too burdensome, in particular for smaller businesses."
He acknowledged that such proposals are "politically sensitive", but added: "Brexit was not our choice. It is the choice of the UK.
"Our proposal limits itself to what is absolutely necessary to avoid a hard border."
Meanwhile, former British foreign secretary Boris Johnson has claimed Downing Street is negotiating a backstop "that makes the UK a permanent EU colony".
This backstop closes off our options. It means we cannot have no deal / WTO or SuperCanada. Under either our rules could differ from the EU’s – so they will keep us in the backstop instead. It means, in fact, that the only possible deal is Chequers 3/4— Boris Johnson (@BorisJohnson) October 10, 2018