Is a new type of TSA pre-approval required before commencing flight training?
(Answer completely rewritten based on new information)

Does the TSA plan to implement rules that require every prospective student to seek pre-approval from a government agent (not just a flight school) prior to beginning flight training?

Not exactly. US citizens can still start flight training whenever they want. But from April 1, 2016 pre-approval from "a government agent" (the TSA) is required prior to soloing. Foreigners will still need TSA approval before starting flight training.



The final rule on this has now been published in the Federal Register and the TSA will indeed vet all student pilots, as of 1 April 2016 (my emphasis):


  This action requires  applicants to apply for a student pilot 
  certificate through a Flight Standards  District Office, designated
  pilot  examiner, airman certification  representative associated with
  a pilot  school, or certified flight instructor.  Aviation Medical
  Examiners will no  longer issue a combination medical  certificate and
  student pilot certificate.  Student pilot certificates will be issued 
  on the same medium as other pilot  certificates and will have no
  expiration  date. All student pilot certificates issued  before the
  effective date of this final rule  will expire according to their
  terms  unless they are replaced by another  pilot certificate. This
  final rule responds  to section 4012 of the Intelligence  Reform and
  Terrorism Prevention Act  and facilitates security vetting by the 
  Transportation Security Administration  of student pilot applicants
  prior to  certificate issuance. This action  withdraws the proposal
  for pilot  certificates to include a photograph of  the individual
  pilot.


But note that a student pilot's license is still not required in order to start training, it's only required in order to solo. So under this new rule, a US citizen can start flight training as usual, but can't solo until he gets TSA approval for his student pilot application:


  IRTPA [the Intelligence Reform and 
  Terrorism Prevention Act] required that security vetting  of all individuals, including
  pilots, must  be successfully completed by TSA  before the FAA issues
  a certificate.  Therefore, applicants for student pilot  certificates
  must be vetted to receive  their certificates and operate an aircraft 
  as pilot in command.


This also means that AMEs will no longer issue student certs, only medical certificates. The FAA expects that processing will take several weeks, which may be an issue for accelerated training programs that aim to get you a full pilot's license in a month or less:


  The FAA estimates that the  turnaround time for student pilot 
  certificates can be reduced to an average  of 3 weeks or less,
  provided that initial  security vetting by TSA indicates that  the
  applicant is eligible for the  certificate.


There was also some concern that it wouldn't be possible for pilots to solo on their 16th birthday any more because of the processing time, but the FAA specifically addressed that point, essentially by asking students to submit their application well in advance.

On the 'plus' side, the new regulation specifically says that pilots licenses will still not have a photograph (at least for now), which would have added more time and costs to the whole process. And the new student certificates will be plastic, not paper.