Esmaili did not identify the woman, but a London-based British Council employee, Aras Amiri, was arrested previous year in Iran during a trip to visit relatives.
Iran does not recognize dual nationality.
Zaghari-Ratcliffe's husband, Richard Ratcliffe, has been separated from his daughter, Gabriella, who was with her mother when she was detained in Iran and has since remained in the country with her grandparents.
Zaghari-Ratcliffe, who works for the charity arm of Thomson Reuters, was arrested in April 2016.
Iranian authorities shut down the British Council more than a decade ago for what Esmaili described as "illegal activities".
The British Council had released a statement earlier this month saying it was aware of the detention of a staffer following the arrest of Iranian national Aras Amiri.
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The British Foreign Office said it was "very concerned" at the reports.
"We have not been able to confirm any further details at this stage and are urgently seeking further information", a British Foreign Office spokeswoman said.
A court sentenced the individual to 10 years in prison after the confessions, Esmaili added. She is serving a five-year sentence for spying - a charge she denied.
The British Council said previous year that it was aware that one of its staff had been detained while making a private family visit, without identifying them as Ms Amiri.
Ms Amiri is being held Tehran's notorious Evin Prison, where Ms Zaghari-Ratcliffe is also an inmate.
The spokesperson said she was "an Iranian student who wanted to live and work in the UK", was hired by the British Council and "repeatedly travelled to Iran under aliases. and made connections with artistic and theatre groups".
Ms Zaghari-Ratcliffe's husband revealed earlier this year that his wife had been asked to spy on the United Kingdom in exchange for her freedom, which she refused.