Johnson will meet with his Iranian counterpart Javad Zarif and will also hold talks on the Iranian nuclear deal.
Nazanin Zaghari-Ratcliffe, a project manager with the Thomson Reuters Foundation, was sentenced to five years in jail after being convicted by an Iranian court of plotting to overthrow the clerical establishment.
She is due to face fresh charges of "spreading propaganda" during a court appearance on Sunday.
Mr Johnson's trip to Iran will be the first visit by a British foreign secretary since 2015 and only the third since 2003.
He will also discuss British concerns over Iranian involvement in conflicts in the Middle East, especially in Syria and Yemen.
"We fervently hope the Foreign Secretary's visit can secure her speedy release, but if it doesn't we'll go on campaigning".
"Johnson has insisted that the second set of charges laid against her are not related to his error last month in telling the foreign affairs select committee in Westminster that she had been in Iran to train journalists", The Guardian says.
TRF chief executive Monique Villa said recently that the new charges risked a further 16-year prison sentence and were "a mockery of justice".
The Foreign Office has always stressed that the payment of the money is in no way linked to the cases of Mrs Zaghari-Ratcliffe and the other imprisoned dual nationals. She was travelling with her daughter Gabriella, now three years old.
Zaghari-Ratcliffe's fate become a major political issue in Britain after Johnson made remarks on November 1 that appeared to cast doubt on statements from her employer about what she had been doing in Iran.
It is understood that Nazanin's husband Richard will not accompany Mr Johnson after receiving advice that it may not help his chances of seeing his wife in prison. In reality, she was visiting family in the country.