The amendments are unlikely to pass, but will send a message to government whips amid claims that more ministers could follow Boris Johnson and David Davis in resigning over Mrs May's plan.
Johnson said the plan would make it much more hard to do trade deals.
"It may resolve the dilemma the Prime Minister faces".
Winning the backing of the DUP for one of the amendments is particularly significant because the Northern Irish unionist party had promised to support May's government in confidence votes and Brexit policy, after the Conservatives lost their overall majority in the 2017 election.
More than 1,000 readers took part in an online survey, answering questions on the Prime Minister's leadership and the prospects for the UK's departure from the EU.
Virendra Sharma, a supporter of the Best for Britain campaign for another referendum, said that a "gang of Brexit bullies are threatening to pull the plug on the Prime Minister".
The Daily Telegraph said on Tuesday night that senior Brexiteers said that they have enough letters in hand to trigger a confidence vote and will submit them unless she hardens her Brexit plans.
Asked if May should remain in Number 10, Trump "offered few words of support" for the beleaguered prime minister, says The Independent. Mr. Johnson was quickly replaced by 51-year-old former Health Minister Jeremy Hunt, who unlike Mr. Johnson supported staying in the European Union in the 2016 referendum.
He said the British position is now "a basis for a real negotiation now and what we haven't had for many months is a clear negotiating position".
The foreign and Brexit secretaries have both quit over the plans. The Conservatives now have 316 MPs so 48 of them would need to write such letters to challenge May.
Tories unhappy with the current Brexit plans need to accept the "harsh truth" that there's no other option, William Hague has said.
Two years after Britain voted 52 per cent to 48 per cent to leave the European Union, May is trying to find a middle way between two starkly differing views - within her party and the country - of the UK's relationship with Europe.
Mrs May has insisted the Chequers plan delivers on the 2016 referendum result, but her Brexiteer critics have argued that will leave Britain tied too closely to European Union rules for the foreseeable future.
Mr Watson said Labour was keeping the option of a second Brexit referendum open in the case that Parliament could not decide a way forward, though he stressed that this route was "highly, highly unlikely".
He had said: "The Government's commitment at Chequers to the political and constitutional integrity of the United Kingdom with no borders between Northern Ireland and the rest of the United Kingdom is a welcome reaffirmation of what is an absolute priority for us".