The Democratic Unionist Party (DUP) does not have the new prime minister Boris Johnson "over the barrel", its deputy leader insisted.
Nigel Dodds MP said the DUP needs to "refresh and renew" their confidence and supply agreement with the "effectively new" government.
The Tories have relied on the DUP's 10 MPs since the 2017 general election.
However, the DUP will want to see Mr Johnson's "plans going forward", Mr Dodds told the BBC.
He reiterated that the DUP's priorities were strengthening the union, delivering Brexit and restoring devolution in Northern Ireland.
Speaking later in the House of Commons, Mr Dodds said that he "warmly welcomes" Mr Johnson's "optimism".
"Let us strain every sinew to strengthen the union, get a deal to leave on the right terms," he added.
Mr Johnson thanked the DUP for its work in stopping Labour from gaining power.
Taoiseach Leo Varadkar has said he hopes Boris Johnson has not chosen to pursue a no-deal Brexit.
The Irish prime minister was speaking during a meeting of the Irish cabinet in County Donegal.
"A no-deal Brexit is a British threat and the only people who can cause a no deal is the UK government," he said.
"Our position has not changed. The backstop is an integral part of the withdrawal agreement and without the backstop there is no withdrawal agreement, there is no transition phase and there is no implementation phase, and there will be no free trade agreement until all those matters are resolved.
"So I am hoping the new British prime minister has not chosen a no deal, but that would be up to them."
Speaking in the Commons on Thursday, former Northern Ireland Secretary Karen Bradley asked the new PM to prioritise the needs of victims of the troubles and survivors of historical institutional abuse.
Thanking her for her contribution, Mr Johnson said he believes there is common ground in the House that military veterans should not face prosecution for alleged crimes when there is no new evidence against them.
Earlier on Thursday, a former top civil servant said that the Ms Bradley's replacement Julian Smith "must move" on same sex marriage and new abortion regulations.
In July, peers backed moves to introduce same sex marriage and abortion reform in Northern Ireland.
Mr Smith, who voted in support of both in Parliament, will need to take action if not quite "hit the ground running", Stephen Grimason said.
Mr Grimason, who served as the Northern Ireland Executive's director of communications for more than 15 years, said he expected the former chief whip to have a "reasonably close relationship with the DUP".
Speaking to the BBC, Mr Grimason added that Sinn Féin was "already shifting nervously in their seat" about its prospects under his tenure.
"Julian Smith, with his background in the whip's office, he's a fixer, he's a nudger, he is a cajoler... he has to make deals every day to get government business through," said Mr Grimason.
Analysis: What challenges face the new NI secretary?
By Gareth Gordon, BBC NI political correspondent
Any new Conservative secretary of state comes to Northern Ireland already loaded down by baggage - at least as far as Sinn Féin are concerned.
It's even more true if they've been welcomed onstage at the DUP conference as a "friend" of that party.
That's what happened when Julian Smith attended the annual conference of the DUP in 2017 following the confidence-and-supply agreement which binds the DUP and the Tories together until - or indeed if - a future general election ends in divorce.
And it's bound to be thrown back at the former Conservative chief whip when he arrives at Stormont House to attempt to steer the talks towards a restoration of devolution with a sizeable dollop of Brexit on the side.
Against that, he was the man with the impossible job of getting enough of his MPs to vote for Theresa May's doomed Withdrawal Agreement.
As chief whip that was his job, but it put him on the opposite side of the argument to the DUP.
So welcome to Northern Ireland, Mr Smith.
At least you don't have a big act to follow.
Your time in Northern Ireland cannot possibly be as unsuccessful as Karen Bradley's. Can it?
Who is Julian Smith?
On Wednesday, outgoing secretary Karen Bradley was sacked by the new prime minister. before the new cabinet was revealed.
Julian Smith has been an MP for Skipton and Ripon since 2010.
Having backed the Remain campaign in the 2016 Brexit referendum, he most recently served Theresa May as chief whip - a job in which he was unable to guide her proposed EU Withdrawal Agreement through the House of Commons.
In that role he also worked closely with the DUP and attended the party's annual conference in 2017.