  The paper studies the quantum action for the four-dimensional real
$\phi^4$-theory in the case of a general formulation using the background field
method. The three-loop renormalization is performed with the usage of a cutoff
regularization in the coordinate representation. The absence of non-local
singular contributions and the correctness of the renormalization
$\mathcal{R}$-operation on the example of separate three-loop diagrams are also
discussed. The explicit form of the first three coefficients for the
renormalization constants and for the $\beta$-function is presented.
Consistency with previously known results is shown.
