The prime minister will chair a conference call on Sunday afternoon to update ministers following a weekend of intense negotiations in Brussels.
No 10 has kept the status of the talks closely guarded, with only Mr Johnson, cabinet secretary Sir Mark Sedwill and a small number of advisers aware of the likelihood of a deal being struck before the 31 October deadline.
As hopes rose that an agreement could be possible, No 10 said the Queen’s Speech would include the announcement of a number of Brexit-related laws designed to “seize the opportunities” of leaving the EU.
These are expected to include plans to speed up access to new medicines and treatments, introduce a points-based immigration system and cut regulation.
The speech, which is used by the government to announce its legislative plan for the coming parliamentary session, will also include bills on health, crime, science and infrastructure among 22 pieces of new legislation.
The priority for new legislation will inevitably be Brexit, however. If a deal with the EU is agreed at the European council summit next week, the government will have just over a week to get it approved by parliament and enacted in law before 31 October. MPs will likely have to sit at the weekend and late into the night to pass the necessary legislation.
Ahead of the speech, Mr Johnson said: “Getting Brexit done by 31 October is absolutely crucial, and we are continuing to work on an exit deal so we can move on to negotiating a future relationship based on free trade and friendly cooperation with our European friends.
“But the people of this country don’t just want us to sort out Brexit. They want their NHS to be stronger, their streets safer, their wifi faster, the air they breathe cleaner, their kids’ schools better-funded – and this optimistic and ambitious Queen’s Speech sets us on a course to make all that happen, and more besides.”
He added: “After one of the least-active parliaments in living memory, the proposals we are bringing forward will get this country moving again. This is a Queen’s Speech that will deliver for every corner of the UK and make this, once again, the greatest place on earth.”
No 10 sought to dampen speculation of a major breakthrough in Brexit talks, saying there was “a long way” to go before an agreement was reached.
A source said: “We’ve always wanted a deal. It is good to see progress, but we will wait to see if this is a genuine breakthrough. We are a long way from a final deal and the weekend and next week remain critical to leaving with a deal on 31 October. We remain prepared to leave without a deal on 31 October.”
Mr Johnson’s hopes of securing parliament’s approval for an agreement were dealt a blow on Saturday when Nigel Dodds, the DUP’s deputy leader, warned that the proposal reportedly being discussed “will not work”.
The plan would see Northern Ireland, but not the rest of the UK, remain in the EU customs union after Brexit. But Mr Dodds told Italian newspaper La Repubblica: “Northern Ireland must stay in a full UK customs union, full stop.
“There is a lot of stuff coming from Brussels, pushed by the Europeans in the last hours, but one thing is sure: Northern Ireland must remain fully part of the UK customs union. And Boris Johnson knows it very well.”
He added: “It cannot work because Northern Ireland has to remain fully part of the UK customs union.”
Mr Johnson announced on Saturday that the Queen’s Speech will also include a new law to tackle air pollution and reduce the use of plastics.
The new Environment Bill will succeed the 1956 Clean Air Act and is expected to include a legally-binding limit for the most dangerous types of air pollution.
However, the exact threshold is likely to be subject to a consultation, meaning it is not clear how tough the new law will be.
The legislation is also expected to force companies to cover the entire cost of collecting and recycling their packaging, while local councils will also be given new duties.
And it will pave the way for the introduction of deposit return schemes for drinks bottles, and the creation of a new Office for Environmental Protection.
Mr Johnson told The Times: “This landmark legislation will set a framework of far-reaching, legally binding targets to reduce plastics, restore biodiversity, improve the quality of our water and clean up our air.”