Efforts to reach a Brexit deal before Thursday's summit of European leaders are continuing in Brussels.
Negotiators from both sides are trying to bridge what senior EU official Michel Barnier called "big gaps".
EU ambassadors were told on Sunday the UK would make concessions on its customs plan for Northern Ireland.
Speaking in the House of Commons, Boris Johnson said the government was preparing to leave on 31 October and it was time to "get Brexit done".
Outlining his legislative agenda for the year ahead - which includes seven Brexit-related bills - the prime minister hit out at those who were advocating what he called more "dither and delay".
Both sides have said they hope to agree a deal before the EU summit on Thursday and Friday, and if that happens, the government says it will introduce a withdrawal agreement bill to be voted on next Saturday in a special Parliamentary session.
Sitting hours of Parliament planned for Saturday are 9.30 til 2.30— Laura Kuenssberg (@bbclaurak) October 14, 2019
The PM's official spokesman told journalists on Monday morning: "Talks remain constructive, but there is a lot of work still to do."
That echoes the message delivered by Mr Johnson to his cabinet on Sunday and the latest comments by Ireland's Tanaiste (deputy prime minister) Simon Coveney
"A deal is possible, and it's possible this month," Mr Coveney said. "It may even be possible this week. But we're not there yet."
If the Commons backs a deal, the PM's spokesman said Mr Johnson would expect MPs to "work around the clock" to pass the necessary legislation so Brexit can happen on schedule at 23:00 GMT on 31 October.
The issue of the Northern Ireland border in post-Brexit arrangements is seen as the key factor in the EU-UK talks.
Mr Johnson submitted new proposals to the EU earlier this month, and its leaders promised to examine them carefully.
However, a number of figures, including Irish Taoiseach (Prime Minister) Leo Varadkar, said they did not form the basis of a deal.
Hope of progress were faint until Mr Johnson and Mr Varadkar met last Thursday and the Irish leader said afterwards their discussions had been "positive" and "sufficient to allow negotiations to resume in Brussels".
Mr Barnier told EU diplomats in a briefing this weekend the UK had dropped its proposals to include an up-front veto for the Stormont Assembly before any new arrangements for Northern Ireland come into force, said BBC Brussels correspondent Adam Fleming.
But he said the UK was still seeking the power for Northern Ireland to leave the arrangements at some point in the future.
According to a note of his meeting with EU ambassadors on Sunday evening, Mr Barnier also said he would be willing to accept Mr Johnson's plan for Northern Ireland to remain part of the UK's customs territory but apply EU customs procedures.
However, he said he could not accept a British proposal to track goods entering Northern Ireland to determine whether they ended up in Ireland.
Adam Fleming said it appeared EU negotiators had "softened" their position by indicating they were prepared to keep talking until Wednesday - the eve of the summit - despite saying previously that a revised deal had to be ready a week in advance.
In a statement, the EU it added that the "intense technical discussions" between officials would continue on Monday before member states were updated on the progress at a meeting in Luxembourg on Tuesday.
The Irish border has been a policy conundrum for a long, long time, but it seems now there has genuinely been a bit of push and pull, and a little bit of movement on both sides.
There are swathes and swathes of technicalities going on here. One cabinet minister, who was briefed by the prime minister on Sunday, even told me they are blind to the detail.
As far as they are concerned, that's a good sign - it means the talks are genuine and negotiators are able to get on with their work without too much political pell-mell.
But while a deal is possible, it is still a massive if.
The politicians' mood has changed very much in the last seven days, particularly since that meeting between Leo Varadkar and Boris Johnson.
And getting a deal is obviously the most straightforward, politically advantageous way for the government to leave at the end of this month and keep Mr Johnson's promise that got him into No 10.
But it doesn't mean the really, really thorny policy questions have disappeared.
Earlier on Monday, Chancellor Sajid Javid announced he intends to hold the Budget on 6 November, insisting it will be "the first after leaving the EU".
But Labour's shadow minister for the Cabinet Office, Jon Trickett, told Today he would be "surprised" if the Budget went ahead as planned as "we have no idea if they are going to get this Brexit proposal through the House or not".
Timeline: What's happening ahead of Brexit deadline?
Monday 14 October - The Commons returns, and the government has outlined its legislative agenda in the Queen's Speech. The speech will then be debated by MPs throughout the week.
Thursday 17 October - Crucial two-day summit of EU leaders begins in Brussels. This is the last such meeting currently scheduled before the Brexit deadline.
Saturday 19 October - Special sitting of Parliament and the date by which the PM must ask the EU for another delay to Brexit under the Benn Act, if no Brexit deal has been approved by MPs and they have not agreed to the UK leaving with no-deal.
Thursday 31 October - Date by which the UK is currently due to leave the EU.