An NRI can become a Director in an Indian Company.
DIN for him needs to be applied first
If he is residing in US and has a US Pass port then duly notorised copy of his Passport ( as identity proof) and an address proof notorised are required to be attached with the DIN application. ( Notorisation should be done in the country of his Residence (i.e. US)
Also his DIN application is required to be notorised in the country of residence.
Be careful while generating DIN , there should be no mismatch in the details given in the DIN application and the proofs attached.
( if the NRI is residing in India then a copy of his VISA should be attached with the passport ) ( In that case notorisation of his proofs with the application can be done in India also)
If you still find problem then contact DIN cell - at this no 95120 -6522478.