Lindsay and Craig Foreman, from East Sussex, were arrested during a motorcycle world tour in January and accused of spying - allegations they deny.
Lindsay's son Joe Bennett said the pair made a court appearance on Saturday and that a verdict was expected within days.
"Although they're mentally resilient, I think their physical health is really starting to suffer," he was quoted as saying by BBC Radio 4. Their family is "exhausted", he added, and "terrified by what may come next".
"I have nightmares thinking about how Mum and Craig can possibly be coping with this injustice, hearing false accusations, feeling helpless to be able to defend themselves, knowing that their words have been mistranslated," he added.
A UK Foreign Office representative was not allowed to attend the couple's Tehran trial on Saturday, the Guardian reported on Thursday, citing their family.
Critics and rights groups have long accused Tehran of engaging in so-called hostage diplomacy by detaining foreigners in exchange for jailed nationals abroad or economic and political concessions.