Part 1: Can a foreigner with a PPL in another country's system receive training in the US?
All other normal prerequisites such as ASFP, medicals, etc. aside, yes, a foreign license holder can receive training in the U.S.
Remember: You do not need a pilot certificate to receive training, but you would need one to solo or take a check ride (such as a student certificate).
If a person holds a foreign certificate, I believe they have to be flying an aircraft registered in their home nation in order to qualify to solo or otherwise act as PIC in a foreign country without obtaining a local certificate.
So, receive training? Sure. Solo/PIC/check-ride in an N-registered? No.
If what you mean to ask is "can he obtain a new cert/rating without converting". No because the FAA does not issue South Korean pilot certificates, so that would not be possible. The dual training received in the U.S. would certainly be able to be logged and count as valid training back home.
Part 2: ASFP Training Categories
From the flightschoolcandidates.gov FAQ section:
What category do I qualify for?
Candidates seeking flight training are separated into one of four
different categories. A brief explanation of the categories follows:
Category 1 - Candidates who seek flight training in the operation of
aircraft with a maximum certificated takeoff weight (MTOW) greater
than 12,500 pounds, but who do not fall into Category 2.
Category 2 - Candidates who seek flight training in the operation of
aircraft with a maximum certificated takeoff weight (MTOW) greater
than 12,500 pounds, and who:
Are employed by a foreign air carrier that operates under 14 CFR part
1546; Have unescorted access authority to a secured area of an airport
under U.S.C 44936(a)(1)(A)(ii),49 CFR 1542.229; Are a flight crew
member who has successfully completed a criminal history records check
in accordance with 49 CFR 1544.230; or Hold an airman's certificate
that is recognized by the FAA or appropriate US military agency, with
a type rating for a multi-engine aircraft that has a certificated
takeoff weight of 12,500 pounds or more.
Category 3 - Candidates who seek flight training in the operation of
aircraft with a maximum certificated takeoff weight of 12,500 pounds
or less for the following training events:
Single Engine Land (SEL)
-- Includes initial Airman's Certificate, including a private, recreational, or sport pilot certificate. If a private and/or
commercial license is the candidate's initial FAA license, it is
considered an initial airman's certificate and is not exempt.
Instrument Rating (IR) Multi Engine Land (MEL) Each of these training
events requires a separate training request. Clarification regarding
Category 3 exemptions is covered in the next Frequently Asked
Question.
Category 4 - Candidates who seek recurrent training in the operation
of aircraft with a maximum certificated takeoff weight (MTOW) greater
than 12,500 pounds, and are current and qualified on the aircraft for
which they are requesting training. These training requests are
submitted by the flight training providers -- Candidates need to
"Select Providers for Recurrent Training" from the AFSP Candidate
account. If you have further questions regarding your category, please
contact your Flight Training Provider.
If he wants training intended for an instrument rating in an aircraft under 12,500 Lbs MTOW then category 3.