While both ministers stressed securing Parliamentary support for PM May's deal, Ms. Rudd did not rule out a "plausible" second referendum to remove an impasse between MPs.
But Mrs. Leadsom called a second referendum "unacceptable", urging MPs to approve a "managed no-deal" instead.
However, Downing Street has dismissed both proposals, with the prime minister's spokesperson flatly responding "no" to Ms. Rudd's idea.
The prime minister has "been very clear on the dangers of calling a second referendum," he said, adding that Ms. Rudd was "clear" to push Mrs. May's deal through Parliament.
"This is not something that is available. The EU has been very clear that there is no withdrawal agreement available that does not include a backstop."
Mrs. Leadsom's Argument
Mrs. Leadsom explained on BBC Radio 4's Today programme that: "A managed no deal does not necessarily mean there is no withdrawal agreement at all," arguing that MPs could shape a stripped-down deal which included some of the EU's no-deal contingencies.
"What I am looking at is trying to find an alternative so that in the event that we cannot agree to this deal that there could be a further deal that looks at a more minimalist approach but enables us to leave with some kind of implementation period," she added.
"That avoids a cliff edge, that avoids uncertainty for businesses and travellers and so on."
When asked about Amber Rudd's "plausible" alternative, Mrs. Leadsom replied: "It's not government policy. I myself think it would undermine the biggest democratic exercise ever, where we had a clear majority to leave the European Union."
"To have a second referendum would unfortunately be going back to people and telling them they have got it wrong and they needed to try again," she continued. "I think it would be unacceptable."
Ms. Rudd's Argument
Ms. Rudd has long supported the Remain camp since 2016 and expressed doubt that Parliament would back the prime minister's Brexit draft plan, instead arguing for a "People's Vote" contingency if MPs rejected Mrs. May's proposal.
"I have said I don't want a People's Vote or referendum in general, but if parliament absolutely failed to reach a consensus, I could see there would be a plausible argument for it," she said on ITV's Robert Peston show.
Good to join @peston. As I made clear I don’t want a people’s vote or a referendum in general. Parliament has to reach a majority on how it’s going to leave the European Union.— Amber Rudd MP (@AmberRuddHR) December 19, 2018
That’s why I ask all colleagues to support @theresa_may. https://t.co/7607EjPJUF
"Parliament has to reach a majority on how it is going to leave the EU," she continued. "If it fails to do so, I can see the argument for taking it back to the people again as much as it would distress many of my colleagues."
The UK will leave the European Union on 29 March 2019 but must agree on withdrawal terms and future relations with Europe, which can only be implemented following approval from UK and EU parliaments.
The prime minister delayed the much-anticipated Commons vote to 21 January, prompting further backlash from MPs. MPs are set to debate her draft plan in Commons for five days starting 9 January. Should Parliament reject her bill, the UK must leave the EU in March, prolong the withdrawal process or rescind triggering Article 50.