The Northern Ireland Secretary of State said there are only a small number of differences between Sinn Fein and the Democratic Unionists, mainly around Irish language rights, culture, identity, and respect.
But the DUP MP Jeffrey Donaldson insisted that the Armed Forces Covenant, which aims to give ex-servicemen and women some priority medical treatment, housing assistance and school places for children, was integral to a deal.
“There will be no outcome that will not see that Armed Forces Covenant providing for the servicemen and women and the veterans and their families from Northern Ireland who have served this country,” he said.
Mr Brokenshire said the British government wanted to see the ex-soldiers’ benefits “touch” all parts of the UK.
The DUP and Sinn Féin failed to meet Mr Brokenshire’s original Monday deadline for a deal. Stormont had been effectively in limbo since January.
The crisis sparked a phone call between British prime minister Theresa May and Taoiseach Leo Varadkar on Wednesday while he was on a trade mission to the US west coast.
“They talked about the responsibility on parties to overcome their differences and the need for them to compromise on the outstanding issues for the benefit of all communities in Northern Ireland,” said Mrs May’s official spokesman.
Mr Brokenshire said a last-minute powersharing deal would be conditional on a budget being agreed and passed by the end of November.
He also said he will reflect carefully on MLAs’ salaries - £49,500 a year - which cannot be stalled or docked without primary legislation in Westminster.
Amid calls from the Alliance deputy leader Stephen Farry and others for a mediator to support the talks, Mr Brokenshire said the Government was considering other interventions and other ways to broker a deal.
But he added: “I think it still remains possible but it is certainly difficult.
“We will continue to keep a range of options available to us to see how we can move this process forward.”
Mr Brokenshire told British MPs he believed the best way forward for dealing with legacy allegations against former servicemen and women for incidents during the Troubles had been agreed at previous talks.