Johnson has been on the defensive over the past week since being forced to correct remarks he made to a British parliamentary committee, when he said that Zaghari-Ratcliffe, who has been in jail since she was detained at Tehran airport in April 2016, had been teaching journalism during her visit to the country.
It comes as another Cabinet minister, Michael Gove, was also criticised for comments about the jailed Briton. On Wednesday, Iranian state TV IRIB said Johnson amounted to an unintentional confession.
Ms Zaghari-Ratcliffe insists she and her young daughter were just on a family visit when she was arrested in Tehran in 2016.
Last month, Iran's Islamic Revolutionary Guard Corps (IRGC) opened a new case against her, accusing her of having joined organizations specifically working to overthrow the regime.
Earlier this month, Johnson told lawmakers that Zaghari-Ratcliffe was "teaching people journalism" when she was detained past year.
Richard Ratcliffe said he spoke to Mr Johnson for about 20 minutes yesterday morning, during which the Cabinet minister said he was "deeply sorry for Nazanin's suffering".
On Sunday, Johnson's colleague Michael Gove, the UK Environment Minister, added yet more uncertainty during an interview with the BBC. Mr Khan, who succeeded Mr Johnson as London mayor, told the Andrew Marr Show: "I think it's important that Boris Johnson clarifies the huge error he made because he's been prayed in aid by Iranian state TV and others to say because of what he said it's an admission on the part of the British Government as to what Nazanin's alleged to have done".
Mr Johnson has agreed to meet Mr Ratcliffe in the next few days, the family's statement said.
New lumps have been found and she has complained of sharp stabbing pains in her breasts for some months, Mr Ratcliffe 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. Told that Mr Ratcliffe said his wife was in Iran on holiday, he added: "In that case, I take exactly her husband's assurance in that regard".
"Theresa May must ensure Nazanin Zaghari-Ratcliffe does not pay the price for her ministers' bungling".
After Mr Gove's remark on the BBC's Andrew Marr Show, he was told her husband's explanation.
British ministers rallied around Boris Johnson on Sunday, saying the foreign secretary was doing a "great job" and had no reason to resign over remarks that critics say may prompt Iran to extend the prison sentence of a jailed aid worker.
"Michael Gove must have known he would be asked that question and yet answered in that way".
"She is not a journalist and has never trained journalists at the Thomson Reuters Foundation, where she is project manager in my Media Development team", Monique Villa, the charity's chief executive, said.
Labour's Chris Bryant said the comments were proof of an "incompetent" Government.
The leader of Britain's main opposition Labour Party, Jeremy Corbyn, called on May to sack Johnson, writing in the Observer newspaper on Sunday that "we've put up with him embarrassing and undermining our country through his incompetence. for long enough".