Normally you would expand about a fixed point where $V$ is extremum but, if not suppose a point particle in 1d as an example:$$L= \tfrac12 m \dot{x}^2 - (V_0+A x + \tfrac12 k x^2)\, .$$The EOM for this is$$m\ddot{x}=-A-kx=-k \left(\frac{A}{k}+x\right)\, , \tag{1}$$with $A\ne 0$. Eq.(1) is a linear ODE so you have linearized the EOM.
Then define $X=x+\frac{A}{k}$ so that$\ddot{x}=\ddot{X}$ and you have the usual harmonic approximation for $X$.The factor $\frac{A}{k}$ just adds a constant shift to $x$, and this shift can usually be ignored unless $V’’=0$. You should be able to easily convert this argument to fields.