For NIH grants the guidance is in the Grants Policy Statement (pg 18 of the
printed version).

Principal Investigator. The PI (who also may be known as the PD) is the
individual, designated by the grantee, responsible for the scientific or
technical aspects of the grant and for day-to-day management of the project
or program. The PI is not required to be an employee of the grantee. However,
because the grant, if awarded, is made to the organization, the applicant
organization must have a formal written agreement with the PI that specifies
an official relationship between the parties even if the relationship does not
involve a salary or other form of remuneration. If the PI is not an employee of
the applicant organization, NIH will assess whether the arrangement will result
in the organization being able to fulfill its responsibilities under the grant, if
awarded.

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