Prior to Tusk's comments, a n European Union official said on Thursday morning that a 20-page declaration had been finalised ahead of an European Union summit on Sunday where Prime Minister Theresa May hopes to have the Brexit divorce deal signed off.
Damian Hinds MP appeared on the breakfast show to discuss Theresa May's claims that she has the backing of the Cabinet on her divisive Brexit deal.
However, the cabinet minister said she still thinks the deal is likely to get through parliament, saying, "I think likelihood is that, despite what people say, the withdrawal agreement will get through".
But Rudd told BBC Radio 4's Today programme: "It is my view that the House of Commons will stop no-deal".
An EU summit is taking place in Brussels on Sunday to "finalise and formalise" the draft deal, and if it is agreed, it will then be voted on in the House of Commons.
He said Mrs May had repeatedly ruled out such an approach but "if she reflects between now and the vote we will look at what she brings back" however the customs concession "wouldn't be the only thing" sought by Labour.
The shadow Brexit secretary, Keir Starmer, has used the same argument as Rudd to make the opposite case - that MPs can safely vote down May's deal because parliament will stop a no-deal exit.
Earlier this week, the Democratic Unionist Party (DUP) vowed to vote down the deal, accusing the government of breaking its promise to the party and Northern Ireland.
United States stocks take sharp losses as tech, internet companies drop
Apple's market value has dropped by $264 billion since early October and Amazon has fallen by $251 billion since early September. The Dow and S&P 500 erased gains for this year while the Nasdaq dropped into correction territory on Tuesday at close.
"As the prime minister said last night, she had a good meeting with Jean-Claude Juncker where further progress was made in the negotiations", her spokesman said.
"After these comments from Amber Rudd, it's time for the government to drop the false choice between a bad deal and no deal, and to come forward with a plan that can command the majority support of Parliament".
The People's Vote campaign for a second referendum said Rudd had "torpedoed the PM's threat of no deal".
(PA Graphics) Mrs May came under pressure at Prime Minister's Questions from Conservative backbenchers who fear her draft agreement will leave the United Kingdom trapped in a customs union with no way of leaving without the EU's consent.
May claimed the refusal to sign up to her deal would create uncertainty and open up the prospect of the United Kingdom remaining in the European Union post-March 29, 2019 - the date set for Brexit.
German Chancellor Angela Merkel set her face against any change to the deal to give the United Kingdom the independent right to end the backstop.
But answering questions in the Commons on Wednesday, she told MPs that the alternative to her deal "will either be more uncertainty, more division, or it could risk no Brexit at all".
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