Zaghari-Ratcliffe's family has been campaigning for her release since she was held in Iran in April 2016 on sedition charges. She is currently under house arrest.
Her husband Richard Ratcliffe wrote on Twitter on Sunday: "This week the Foreign Office warned us of threats to Nazanin and her family if we continued to publicise Nazanin's release date."
He retorted that the family believes "transparency is the best form of protection from abuse."
Unless Iran finds new grounds to extend her detention, Nazanin's official release date is March 7, according to Ratcliffe.
Zaghari-Ratcliffe worked for the Thomson Reuters Foundation, the media organisation's philanthropic arm.
'Mafia tactics'
She denied charges of sedition but was convicted and jailed for five years. She has spent more than four years in jail or under house arrest.
Foreign Secretary Dominic Raab said last month that the March 7 date was "based on the existing sentence," stressing that Iran had repeatedly dashed hopes for Zaghari-Ratcliffe's release after postponing another trial in November.
Britain is "pushing as hard as we can to get the immediate release, not in seven weeks, but as soon as possible, of Nazanin and all of our other dual nationals", Raab said on January 17.
Ratcliffe said on Sunday that the British government's role is to remind Iran that Nazanin "has the UK's protection".
Britain should not "act as a messenger for IRGC (Islamic Revolutionary Guard Corps) mafia tactics and suppression," he added.
"If anything happens to Nazanin or her family, or if she is not released to the UK on 7 March - there should be consequences. We will be discussing with the Foreign Secretary his back up plan."
The Guardian on Monday reported that Ratcliffe sent the Foreign Office a note accusing it of a "remarkable lack of judgement".
The Foreign Office did not immediately respond to a request for comment.
Former British foreign secretary Jeremy Hunt tweeted: "This is deeply concerning. I know @DominicRaab and @BorisJohnson will be doing all they can but Nazanin's voice must never be silenced."
Iran has in recent years repeatedly detained foreigners and dual nationals on charges campaigners and governments say are unfounded, with the prisoners only going free after months and sometimes even years of painstaking negotiation.
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