Mrs Zaghari-Ratcliffe is serving a five-year sentence over allegations of spying, which she denies.
In wide-ranging talks with Iranian foreign minister Mohammad Javad Zarif, Mr Johnson will seek to shore up bilateral relations and urge Tehran to stick by the terms of its 2015 nuclear deal.
The visit to Tehran, the first by Johnson and the third by a British foreign secretary since 2003, follows a thaw in relations after the deal lifted worldwide sanctions against Tehran in January 2016.
Ahead of the visit, Mr Johnson had spoken of his desire to secure the release of Mrs Zaghari-Ratcliffe and other dual British-Iranian nationals being held in Iran.
He later acknowledged his gaffe and has repeatedly said this was not the case, but he has been accused of making her situation worse.
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Mr Ratcliffe told the newspaper that Mr Johnson will attempt to visit his wife in jail and meet the head of the Iranian judiciary.
"We will also discuss our bilateral relationship and I will stress my grave concerns about our dual national consular cases and press for their release where there are humanitarian grounds to do so".
He indicated though that the visit will be dominated by more geopolitical issues, including how we can find a political solution to the devastating conflict in Yemen and secure greater humanitarian access to ease the vast suffering there.
In a statement last month, the Iranian embassy in London said: "While being in prison is not a pleasant experience, Mrs. Zaghari has full access to medical care and services, to social worker and has the right to regularly - twice a week - contact her husband, meet her family including her parents who live in Iran and her daughter".
He said: 'Iran is a significant country in a strategically important, but volatile and unstable, region which matters to the UK's security and prosperity.
Mr Johnson said he wanted the visit to be "constructive", however he admitted that "on many issues we will not agree" despite improvements to relations in recent years.
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While in Iran, the British Foreign Secretary will also discuss issues of mutual concern with Tehran regarding the Middle East and the progress of Iran's nuclear agreement.
"But I am clear that dialogue is the key to managing our differences and, where possible, making progress on issues that really matter, even under hard conditions".
She had an appeal over her five-year sentence rejected by the Iranian Revolutionary Court earlier this year, while she is due to appear in court on Sunday on a second charge of spreading propaganda. Mr Ratcliffe had hoped to accompany Mr Johnson on the trip to see his wife for the first time in 19 months but this had been impossible to achieve.
"I think both sides know that this issue of dual nationals being arrested and then locked up in Iran is a long running problem between the two countries".
"Obviously, I would love to go and see them but what would be better than that would be for him to be able to bring them home for Christmas".
The Foreign Office would not confirm the names or number of other people being held in Iran, saying their families had asked for their cases to be kept out of the public domain.
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