Consider a Euclidean path integral say in a real scalar field theory.

$$

\int d[\phi]\exp(-I[\phi])

$$

In the semiclassical approximation we consider stationary points of the action, and expand around them. Now, consider I want to make a semiclassical expansion of the generating functional

$$

Z[J]=\int d[\phi]\exp\bigg(-I[\phi]-\int d^4x\,J\phi\bigg)

$$

I have a doubt, should I consider saddles of $I$ of all the sourced action?

$$

I_J[\phi]\equiv I[\phi]+\int d^4x\,J\phi

$$

Naively i would guess that I gotta take the saddles of the whole exponent, but

my biggest concern then is that if I take saddles of the sourced action, the stationary field configurations will in general have $J$ dependence, and thus after expanding the action around these stationary points $\phi_s$, taking functional derivatives of $Z$ with respect to $J$ will be very dirty since I will have $J$ dependence in every place I have a $\phi_s$.

So, saddles of the sourced or the unsourced action?