Writing in the Sun on Sunday and the Sunday Express, the Prime Minister said: "We are leaving in 25 days. We can do it with a deal if the EU is willing," the BBC reported.
Johnson said his untested plan to use technology to eliminate customs border checks would take the UK out of EU trade rules while respecting the Northern Ireland peace process.
"I say to our European friends: grasp the opportunity our new proposal provides. Join us at the negotiating table in a spirit of compromise and co-operation," he said.
He claimed MPs from "every wing of the Conservative Party", Northern Ireland's Democratic Unionist Party and from Labour have said "our proposed deal looks like one they can get behind".
But he said "there will be no more dither and delay" and the UK will leave the EU on October 31 with or without a deal.
Johnson did not explain how the government would comply with a law passed by MPs which forces the Prime Minister to seek an extension to the Brexit deadline if no agreement has been made by October 19.
In court documents, the government has said the Prime Minister will request a delay as the law requires, despite his public and Parliamentary statements.
Meanwhile, Brexit Secretary Stephen Barclay, writing in the Sunday Telegraph, said that there were "positive sounds" of support from MPs but "we need that noise to grow louder" as the EU summit approaches, reports the BBC.
But he said the UK's negotiation had been "severely hampered" by "hardcore opposition MPs", who voted to prevent a no-deal Brexit.
"My message to the EU and anti-Brexit opposition is this: we are not backing down," he said.
Also on Saturday, Ireland's Prime Minister Leo Varadkar said that "a deal is still possible" but the EU does not believe the current proposals from Mr Johnson "form the basis for deeper negotiations".
The EU is concerned that the UK wants to leave too many details about customs and regulatory checks to be agreed during the transition period after Brexit.
( With inputs from IANS )