The three argue they should be entitled to the same treatment as interpreters with British forces in the Iraq war who were given exceptional leave to remain in Britain and financial help.
The interpreters say they face the threat of being attacked by the Taliban in their homeland because of their work with foreign forces.
Lawyers have lodged proceedings against British Foreign Secretary William Hague and Defence Secretary Philip Hammond at the High Court on their behalf.
The law firm Leigh Day says one of the three Afghans they are representing, named only as Abdul, remains in Afghanistan, where he and his family have been receiving threats by text message.
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"The recent threats made against Abdul and his family further underline the very real dangers these men and their families face as a direct result of their work, and incredible bravery, in support of the British forces in Afghanistan," Rosa Curling from the firm said.
"The government has a duty to ensure that they are not left exposed to the very real dangers posed by the Taliban.
Prime Minister David Cameron has said that Afghan interpreters should only be allowed to stay in Britain "in extremis".
"I do think that when we think of all that we have spent and all the cost in money and human lives we have put into Afghanistan, we should do everything we can to encourage talented Afghans to stay in their country and contribute to it," he added.
Britain is set to withdraw 3,800 of the country's 9,000 troops from Afghanistan this year, as the NATO-led foreign force prepares to withdraw all combat troops by the end of next year.