The Times quoted a "Brussels source" in their story yesterday, but Davis told BBC television that "they sort of made that up."
"I'm not going to do an actual number on air, it would be ridiculous to do that, but we have a fairly clear idea where we're going on this," he said.
"The last time we went through line by line and challenged quite a lot of the legal basis of these things and we'll continue to do that."
British Prime Minister Theresa May on Friday revealed plans for a transition period of around two years after it officially leaves in 2019, during which time Britain would still largely be under current EU rules.
Davis stressed that Britain "would not under any circumstances" accept the supremacy of the European Court of Justice after the transition period, a key issue for Brexit hardliners.
He added that it was "quite likely" that a joint system of EU-UK courts will be agreed in order to resolve disputes.
May's cabinet is divided over certain key issues of Brexit, particularly on what terms Britain wishes to access the EU's single market after it has left.
Johnson heightened tensions last week with a newspaper article spelling out his vision and wrote again in the latest edition of the Sunday Telegraph to lay down his demands.
He said Britain should not adopt any new EU rules made during the transition period and wants London to be able to sign trade deals with countries outside the EU during the two-year period.
Davis denied Johnson's article had changed the content of May's speech, saying the "policy in the prime minister's speech had been coming for a long time.
"I don't think there's been any change of policy in the last few weeks.