Leading judges last week rejected the case brought by businesswoman Gina Miller, who previously fought a successful legal action against the Government over the triggering of the Article 50 process to start the Brexit countdown.
Lord Chief Justice Lord Burnett, Master of the Rolls Sir Terence Etherton and President of the Queen's Bench Division Dame Victoria Sharp will give written reasons for their ruling at a brief hearing in London on Wednesday.
Ms Miller's case is one of three legal challenges against Mr Johnson's controversial decision to prorogue Parliament, which are expected to be heard together by the UK's highest court next week.
The Supreme Court in London is due to hear an appeal by Ms Miller on September 17, and it is expected that separate legal challenges brought in Edinburgh and Belfast will be heard at the same time.
Giving the High Court's decision last Friday, Lord Burnett said: "We have concluded that, whilst we should grant permission to apply for judicial review, the claim must be dismissed."
Ms Miller's barrister, Lord Pannick QC, had argued that Mr Johnson's advice to the Queen to suspend Parliament for five weeks was an "unlawful abuse of power".
Her case was backed by former prime minister Sir John Major and shadow attorney general Baroness Chakrabarti.
Lawyers representing Mr Johnson said Ms Miller's claim was "academic" and urged the court to reject it.
Speaking outside the Royal Courts of Justice after the ruling, Ms Miller said she was "very disappointed with the judgment".
"To give up now would be a dereliction of our responsibility. We need to protect our institutions," she said.
"It is not right that they should be shut down or bullied, especially at this most momentous time in history.
"My legal team and I will not give up our fight for democracy."
The court will deliver its judgment at noon on Wednesday.
Meanwhile. a decision on a Scottish court challenge to the prorogation of Parliament is due to take place on Wednesday.
A group of around 70 parliamentarians had appealed against a ruling by a judge at Scotland's highest civil court that Mr Johnson's suspension of Parliament is lawful.
Judge Lord Doherty dismissed a challenge against the planned prorogation at the Court of Session last Wednesday, saying it is for politicians and not the courts to decide.
A decision on the appeal is expected on Wednesday morning at the Inner House of the court.