I assume that your FAA private license was issued under 14 CFR 61.75, and that section also includes the requirements for adding instrument privileges:
(d) Instrument ratings issued. A person who holds an instrument rating
on the foreign pilot license issued by a contracting State to the
Convention on International Civil Aviation may be issued an instrument
rating on a U.S. pilot certificate provided:
(1) The person's foreign pilot license authorizes instrument
(2) Within 24 months preceding the month in which the person applies
for the instrument rating, the person passes the appropriate knowledge
(3) The person is able to read, speak, write, and understand the
English language. If the applicant is unable to meet one of these
requirements due to medical reasons, then the Administrator may place
such operating limitations on that applicant's pilot certificate as
are necessary for the safe operation of the aircraft.
The "appropriate knowledge test" in this case is the Instrument Rating - Foreign Pilot test, it's described here with a few sample questions. Some more information is here, including the point that you don't need any prior authorization to take the test:
• Requires NO instructor endorsement or other form of written
I'm not sure about the exact process involved (I guess you just do the test and send the certificate to the FAA) but the FSDO that validated your foreign license for VFR should be able to give you all the details.