Boris Johnson has said he “won’t be deterred by anybody” from leaving the EU on 31 October.
The prime minister said he was “cautiously optimistic” of getting a Brexit deal, but the UK would leave by the deadline “whatever happens”.
EU chief Brexit negotiator Michel Barnier said he did not have “reasons to be optimistic” over getting a deal.
Mr Johnson will meet him and European Commission President Jean-Claude Juncker on Monday for talks.
The PM’s comments come after Parliament passed a law forcing him to ask for an extension to Brexit.
Mr Johnson will have to write to the EU on 19 October to ask for an extra three months, unless he returns with a deal – then approved by MPs – or gets the Commons to back a no-deal Brexit.
But despite the new law, Mr Johnson said he would rather be “dead in a ditch” than ask for an extension.
The Speaker of the Commons, John Bercow, vowed to act with “creativity” if Mr Johnson ignored the law, saying it would be a “terrible example to set to the rest of society”.
MPs managed to pass the law before Parliament was suspended – or prorogued – in the early hours of Tuesday morning until 14 October.
Mr Johnson said the government had made the move so it could hold a Queen’s Speech and put forward its new domestic policy agenda.
But opposition MPs claim it was to stop scrutiny in Parliament of his Brexit plans.
Earlier this week, a Scottish court ruled the prorogation was unlawful as it was motivated by an “improper purpose of stymieing Parliament”.
The government is appealing against the decision and a ruling will be made by the Supreme Court in London on Tuesday.
During Mr Johnson’s speech at the Convention of the North in Rotherham, the PM was heckled by an audience member, telling him to “get back to Parliament” and “sort out the mess that you have created”.
Answering questions after his speech, Mr Johnson said: “We are working incredibly hard to get a deal. There is the rough shape of the deal to be done.
“I have been to talk to various other EU leaders, particularly in Germany, in France and in Ireland, where we made a good deal of progress.
“I’m seeing [Mr Juncker and Mr Barnier] on Monday and we will talk about the ideas that we’ve been working on and we will see where we get.”
He added: “I would say I’m cautiously optimistic.”
MPs are still demanding Parliament be recalled to scrutinise a number of Brexit-related issues, including the release of so-called “Yellowhammer” papers – a government assessment of a reasonable worst-case scenario in the event of the UK leaving the EU without a deal.
But Mr Johnson said that “whatever the shenanigans that may be going on at Westminster”, the government would “get on with delivering our agenda and preparing to take this country out of the EU on 31 October”.
He added that there would still be “ample time” for MPs to scrutinise any deal reached with the EU, adding that he “very much hoped” to agree one at the EU summit on 17 and 18 October.
The Times newspaper reported that a Brexit deal could be on the horizon as the Northern Irish Democratic Unionists (DUP) – the party which has a confidence and supply deal with the Conservatives – had reportedly agreed to “shift its red lines” over the backstop.
The backstop is the policy in the existing withdrawal agreement – negotiated between former Prime Minister Theresa May and the EU – aimed at preventing a hard border returning to the island of Ireland, but it has proved controversial with a number of pro-Brexit MPs.
However, the reports were rejected by the DUP’s leader Arlene Foster, who tweeted: “Anonymous sources lead to nonsense stories.”
A UK spokesman in Brussels revealed the negotiating team had “presented some ideas” on an all-island Sanitary and Phyto-Sanitary solution – essentially keeping Northern Ireland aligned with EU regulations on animal and food health.
But the DUP has said it would not support any arrangement that could see Northern Ireland treated differently than the rest of the UK after Brexit.