He said the UK has made a "big concession" in its proposal and asked the EU to give detail on "what they think the problems are" with his plan.
"It's time for us to get together and really thrash this thing out. And I think we can," the Prime Minister said.
He again claimed the UK will leave the EU on October 31, despite a law designed to force him to request an extension to the deadline.
"We will respect the law," the Prime Minister said, but adding: "We will leave the European Union on October 31."
His comments follow a phone call with Emmanuel Macron, in which the French President reportedly told him the EU will decide at the end of the week whether a Brexit deal is going to be possible.
An Elysee official told the BBC: “The President told [Mr Johnson] that the negotiations should continue swiftly with Michel Barnier’s team in coming days, in order to evaluate at the end of the week whether a deal is possible that respects European Union principles.”
However, Mr Johnson told Emmanuel Macron that Brussels should not be lured into the “mistaken belief” that Britain could extend its membership of the bloc beyond October 31.
Downing Street said reports from France of a Friday deadline to have a Brexit deal in place were not "helpful".
The PM's spokesperson said the Government wanted to see the EU "match the compromises the UK has made".
Downing Street said Mr Johnson made clear in calls to European leaders over the weekend that it was the final chance to strike a deal, but said the EU must match the compromises the UK has made.
The PM said he wanted to reach an agreement that is acceptable to both sides – and claimed that his new proposals commanded the support of MPs.
Number 10 warned Brussels that it would be an “historic misunderstanding” to believe the so-called Benn Act could prevent a no-deal Brexit – despite being designed to do so.
A senior Downing Street source said: “This is the chance to get a deal done: a deal that is backed by parliamentarians and a deal which involves compromise on all sides.
“The UK has made a big, important offer but it’s time for the Commission to show a willingness to compromise too.
"If not the UK will leave with no deal.
“The surrender act and its authors are undermining negotiations, but if EU leaders are betting that it will prevent no-deal, that would be a historic misunderstanding.”
Health Secretary Matt Hancock refused to directly answer whether he believes there is a way around the Benn Act, but when asked, he reiterated the government message of Brexit on October 31, deal or no deal.
He said the UK has "made significant compromises" in its recent Brexit offer and claimed "a deal is possible".
But he said that can only be achieved if the EU makes "compromises too so that we can deliver on this and then we can move forward".
Mr Johnson is to make a series of phone calls to other European leaders on Monday, and could travel to the continent later in the week in a bid to secure an agreement.
It is understood that the Government will consider publishing the full legal text, which has so far only been shared confidentially with Brussels, if it is deemed helpful to progress the negotiations.
Despite Mr Johnson urging Brussels to back his proposals, the EU’s chief Brexit negotiator Michel Barnier poured scorn on the chances of the new proposal succeeding, reportedly telling an event in Paris: “If they do not change, I do not believe, on the basis of the mandate I have been given by the EU27, that we can advance.”
In comments reported by The Observer, he also reiterated the EU’s claim that a no-deal outcome would “never be Europe’s choice… it would always be the UK’s choice, not ours”.
Finnish prime minister Antti Rinne told the Welt am Sonntag newspaper in Germany after speaking to the PM that Mr Johnson now understood “what a big mess this is”.
“That’s why I fear that the October summit would be more about extension than concrete solutions,” he said.