The paper studies the $L^p$-approximate controllability problem in time $T$ for a class of semilinear parabolic systems in a bounded domain $\Omega$ in $\bbfR^n$. Considered are two types of problems: $({\cal P}_D)$ and $({\cal P}_N)$. In each problem, the control functions are fairly general, given by the form $v= v(x,t)$. In $({\cal P}_D)$, the control $v(x,t)$ and the nonlinear term $f(y)$, $y$ being the state of the system, enter the equation under a homogeneous boundary condition of the Dirichlet type, and $v$, supposed to be nonnegative on $\Omega\times (0,T)$, belongs to a dense subset ${\cal U}$ of $L^p_+(\Omega\times (0,T))$. In $({\cal P}_N)$, $v$ and $f$ appear on the boundary condition of the Neumann type, and $v$ belong to a dense subset ${\cal U}$ of $L^p(\partial\Omega\times (0, T))$. For example, $({\cal P}_N)$ is written as $${\partial y\over\partial t}-\Delta y= 0,\quad {\partial y\over\partial\nu}+ f(y)\bigl|_{\partial\Omega}= v,\quad y(\cdot, 0)= y_0.$$ In both cases, the problem in the linear case where $f(y)= 0$ is first solved. Based on this result, the semilinear problem is solved as a perturbation of a linear problem cancelling the nonlinear term. The problem is also studied via the Kakutani fixed-point theorem for other semilinear parabolic systems including multivalued coefficients (a case of flux boundary controls).