The British prime minister is due to meet the European Council president at the G7 summit in France later today.
It is understood he will tell Tusk that under no deal, the UK would only owe about £9bn of the £39bn liability agreed by former Prime Minister Theresa May, Sky News reported.
The approach is likely to stoke tension with other EU leaders at the meeting in Biarritz.
In June, a source close to French President Emmanuel Macron, who is hosting the G7 summit, said refusing to pay was the "equivalent to a sovereign debt default".
Others argue the UK is legally bound to pay the £39bn sum and warn that reneging on the obligation will prevent a future trade deal being struck and could even lead to the EU pursuing the funds through the courts.
It follows a war of words between Johnson and Tusk on Saturday over who would be to blame if the UK left the EU without an agreement.
The British government has also ramped up its preparations for Brexit, announcing that an online "60-second checker" will be brought in so businesses and the public can "work out what, if anything, they need to do before 31 October".
Writing in The Sunday Telegraph, Business Secretary Andrea Leadsom said the government would shortly begin an "engagement campaign to get the country and business ready for Brexit".
An education program delivered through free online seminars will also be launched this week to teach businesses "how to import and export after Brexit" and how to "prepare for things like customs declarations and tariffs".
It is still unclear whether MPs opposed to Johnson's Brexit strategy will be able to block no deal.
The prime minister has repeatedly refused to rule out suspending parliament in order stop backbenchers interfering with his plans.
On Sunday, The Observer newspaper reported that Johnson had asked for legal advice on whether parliament could be shut down for five weeks in September.
Quoting leaked emails, the paper said the prime minister had been told that such a strategy may be possible.
But a government spokesperson said: "The claim that the government is considering proroguing parliament in September in order to stop MPs debating Brexit is entirely false."
Labor's shadow Brexit secretary Sir Keir Starmer tweeted that suspending parliament was "outrageous" and that "MPs must take the earliest opportunity to take Johnson on and defeat him".