Prime Minister said his plan would be a “compromise by the UK”, but he hoped the EU would “understand that and compromise in their turn”.
The European Commission said they will “examine proposals objectively”.
The UK is set to leave the EU on 31 October.
The government has insisted it will not negotiate a further delay beyond the Halloween deadline, saying this would be unnecessary and costly for the UK.
However, under the terms of a law passed by Parliament last month, the PM faces having to request another extension unless MPs back the terms of withdrawal by 19 October – two days after a summit of European leaders.
The European Commission’s President Jean-Claude Juncker and Mr Johnson will speak on the phone later, and the two sides’ negotiating teams will also meet.
Mr Johnson said no-deal was not an outcome the government was seeking, but “it is an outcome for which we are ready”.
Boris Johnson told a conference fringe meeting, hosted by the DUP, that he hoped to reach a deal with the EU over the course of “the next few days”.
The issue of the Irish border and how to keep it free from border checks when it becomes the frontier between the UK and the EU has been a key sticking point in Brexit negotiations.
Mr Johnson has said the solution reached by the EU and Theresa May, the backstop, is “anti-democratic” and “inconsistent with the sovereignty of the UK“, claiming it offered no means for the UK to unilaterally exit and no say for the people of Northern Ireland over the rules that would apply there.
The PM used his speech to confirm parts of his offer to the EU.
He said that “under no circumstances” would there be checks at or near the border in Northern Ireland and the proposals would respect the peace process.
It included promising “a process of renewable democratic consent” for the Stormont Assembly on its relationship with the EU going forward.
He also referred to the use of technological solutions to ensure there was no hard border on the island of Ireland.
He said he did not want a deal to be out of reach “because of what is essentially a technical discussion of the exact nature of future customs checks when that technology is improving the whole time”.
Boris Johnson also said he would “protect the existing regulatory arrangements for farmers and other businesses on both sides of the border”.
He added: “At the same time we will allow the UK whole and entire to withdraw from the EU, with control of our own trade policy from the start.”
The PM said this would “protect the union”.
Echoing the main slogan of the conference, Mr Johnson said: “Let’s get Brexit done on 31 October…to answer the cry of those 17.4 million who voted for Brexit and for those millions who may have voted Remain, but are first and foremost democrats and accept the result of the referendum.”
He said the Tories were “not an anti-European party” and the UK is “not an anti-European country”.
The PM added: “We love Europe. We are European.
“But after 45 years of really dramatic constitutional change, we must have a new relationship with the EU.”
Brexit deadline:Prime Minister want Brexit to be done on 31 of this month.