gfortran not finding pre-compiled module despite -I flag

Question

I am trying to create a folder that contains all the modules I may use. Rather than compiling those modules whenever I need them, I pre-compiled the modules within that folder and would like to use the -I compiler flag to point to the pre-compiled modules (referenced in the gfortran manual). Despite that, I get a compiler (linker?) error that the module cannot be found. For example, consider this directory structure:

project_folder/
├─ src/
│  ├─ main.f90
├─ modules/
│  ├─ my_mod.f90

main.f90:

program main
    use my_mod, only : f
    implicit none

    print *, f(20)

end program main

my_mod.f90:

module my_mod
    implicit none
    private
    public :: f

contains
    function f(arg) result(retval)
        integer, intent(in) :: arg
        integer :: retval
        
        retval = arg**2
        
    end function f
end module my_mod

Within the modules directory I compiled my_mod using

gfortran -c my_mod.f90

And within the src directory I tried to compile main using

gfortran -I../modules main.f90

When I try the src compilation I get the following error:

/usr/bin/ld: /tmp/cc82F9xs.o: in function `MAIN__':
main.f90:(.text+0x51): undefined reference to `__my_mod_MOD_f'
collect2: error: ld returned 1 exit status

How can I properly tell gfortran where the modules are so I can use pre-compiled modules?