Tulip Siddiq told Sky News that Nazanin Zaghari-Ratcliffe was “inconsolable”, “sobbing down the phone” to her husband and “couldn’t believe (Mr Johnson) could make comments that would endanger her life”.
Her family says her health is also increasingly precarious: that her mood is “uncontrollable”, she feels “on the verge of a nervous breakdown” and that doctors have found new lumps in her breasts.
It comes as another Cabinet minister, Michael Gove, was also criticised for comments about the jailed Briton.
Pressure on Mr Johnson has been building after he told a parliamentary committee last week that Ms Zaghari-Ratcliffe was “simply teaching people journalism” in Iran.
Ms Zaghari-Ratcliffe insists she and her young daughter were just on a family visit when she was arrested in Tehran in 2016.
Iran’s Revolutionary Guard claims the 38-year-old was plotting to overthrow the regime and the Briton has been denied consular assistance.
Judicial authorities in Iran seized upon Mr Johnson’s comment.
They hauled her in front of a court last week to confirm she was facing new charges of “propaganda against the regime” and could see her original five-year sentence doubled.
Mr Johnson was forced to call Iran’s foreign minister to ensure his remarks was “not misrepresented”.
He later sought to clarify his remarks, telling the Commons the “UK Government has no doubt she was on holiday”.
Mr Johnson said his previous comments were trying to make the point that he “disagreed with the Iranian view that training journalists was a crime”.
On Sunday, he also called Nazanin Zaghari-Ratcliffe’s husband, Richard Ratcliffe, and the family said the 20-minute conversation had been “positive”.
Mr Ratcliffe said he did not think the Foreign Secretary should quit because he had a “crucial role” in fighting for his wife and that “more instability” would not be in her interests.
Mr Johnson has agreed to meet Mr Ratcliffe in the next few days, the family’s statement said.
He will also “look seriously” at the prospect of him accompanying the Foreign Secretary to Iran, and “consider personally” Mrs Zaghari-Ratcliffe’s eligibility for diplomatic protection.
The statement also said she had been prescribed medication for the lumps in her breasts after experiencing “sharp stabbing pains” for several months.
A specialist who examined her believes they are benign but should be monitored because Mrs Zaghari-Ratcliffe has a family history of breast cancer and that the stress she is under could be a contributory factor.
Shortly after the Foreign Secretary spoke to Mr Ratcliffe, Environment Secretary Michael Gove also appeared badly informed about the case in a television interview.
He was asked by the BBC what the Briton was doing in Iran, and replied: “I don’t know”.
Mr Gove was then told her husband’s explanation for Ms Zaghari-Ratcliffe’s visit.
“In that case I take exactly her husband’s assurance in that regard,” he told the Andrew Marr Show.
“Her husband said she was there on holiday, and her husband is the person who should know.”
Defending his Cabinet colleague who is facing calls to resign, Mr Gove also criticised those trying to “blame politicians in a democracy who are trying to do the right thing”.
Labour’s Chris Bryant said the comments were proof of an “incompetent” Government.
He told Sky News: “Boris got into trouble because he hadn’t bothered to check details before he went to the Foreign Affairs select committee.
“Michael Gove must have known he would be asked that question and yet answered in that way. It’s a Government of incompetence”.
Labour leader Jeremy Corbyn has also launched a furious attack on Mr Johnson, calling on him to quit for “putting our citizens at risk”.