Boris Johnson says the government is offering the EU "very constructive and far-reaching proposals" to break the Brexit impasse.
Speaking to the BBC's Laura Kuenssberg, he confirmed the UK's plan would include some customs checks on the island of Ireland after Brexit.
But they would be "absolutely minimal" and "won't involve new infrastructure".
The EU says it has "not received any proposals from the UK" yet that could replace the backstop.
The UK is set to leave the EU on 31 October.
Mr Johnson has said the exit will go ahead with or without a deal - despite MPs passing a law last month forcing him to ask for an extension from the EU if Parliament hasn't voted in favour of a specific deal or leaving without one.
The border between Ireland and Northern Ireland has been a contentious part of the Brexit negotiations since day one.
At present there are no checks on goods moving across it and the backstop was agreed between former PM Theresa May and the EU as an insurance policy to make sure that does not change - and that no infrastructure like cameras or security posts can be installed in the future.
If used, the backstop would keep the UK in a very close relationship with the EU until a trade deal permanently avoiding the need for checks was agreed.
However, the government says it is "undemocratic" and unacceptable.
Speaking on day three of the Conservative conference, Mr Johnson said he believed the UK was offering enough to win the EU round and more detail would be made public soon.
"Yes, I absolutely do," he insisted.
"So, with great respect to all those who are currently anxious about it - and particularly in Ireland - we do think that our proposals are good and creative.
"But I accept also... there may be hard yards ahead."
He added: "That is going to be where the argument is going to be - and that's where the negotiations will be tough."
The PM said there would "have to be a system for customs checks away from the border".
"If the EU is going to insist on customs checks... then we will have to accept that reality," he added.
When it was put to him that it was not the EU who were insisting on customs checks, the PM replied: "Well, let's see where we get to. And as you know, we made some very constructive and far-reaching proposals."
The Irish Taoiseach, Leo Varadkar, said in the event of a no-deal Brexit, there would need to be checks at ports, airports and perhaps at the border.
But he said that would only be the case if the UK left without an agreement, telling the Dail: "We've never been in the position of signing up to checks as part of a deal."
Mr Johnson told Laura Kuenssberg he always knew "things would get choppy" in the lead up to the Brexit deadline.
But the PM believed "fevers would cool" and "tempers would come down" once that moment had passed.
"There's no way of getting Brexit done without... displeasing people who don't want Brexit to get done," he said.
"[There is] no way of delivering Brexit sort of 52% Brexit and 48% Remain - that's just logically impossible."
Mr Johnson added: "I think once we get it done, and once we can begin building a new partnership with our new friends... we can start thinking about how we can do things differently."
At a reception hosted by the DUP at the Conservative party conference in Manchester on Tuesday night, Mr Johnson said the UK had made progress in the negotiations, adding: "I hope very much in the next few days we are going to get there."
The prime minster said that Northern Ireland should remain part of the United Kingdom "forever".
The government has made a number of policy announcements at the conference, from raising the National Living Wage over the next five years to toughening prison sentences for the worst offenders.
But the plans have been overshadowed by allegations that Mr Johnson squeezed the thigh of journalist Charlotte Edwardes under a table at a lunch in 1999.
Mr Johnson has repeatedly denied the incident, telling the BBC: "It's simply not true."
He would not answer whether he thought Ms Edwardes - who has stood by her claims - lied or whether he remembered the lunch.
And while the PM said such allegations should be taken seriously, he did not agree to an investigation, saying he wanted to "get on with delivering on... [the] important issue of our domestic agenda".