Lawyers from Leigh Day and Co say civilian intelligence officers who give information to the US may be liable as "secondary parties to murder".
The Foreign Office said it did not comment on ongoing legal proceedings.
The lawyers, which include some from the international charity Reprieve, want to establish what official UK policy or guidance is with regard to assisting the US in such cases.
Leigh Day and Co says Mr Khan's father Malik Daud was part of a council of elders holding a meeting in the tribal areas of northwest Pakistan, when a drone missile hit the group.
The firm said it had "credible, unchallenged" evidence Mr Hague oversaw a policy of passing British intelligence to US forces planning attacks against militants.
'Breach of sovereignty'
It will also point out that Pakistan is not involved in an international conflict.