This paper is focused on the feasibility of a neural network with linearly approximated functions on modern FPGA. An approximate multiplier and linearly approximated activation functions were used for a neural network implemented on Zynq FPGA. The neural model was created and learned in the TensorFlow framework. The model parameters were extracted from the Keras model by Python scripts, including the generation of the hardware-accurate model in C++. We proposed a novel architecture for a fully functional, layered, and configurable neural network. The Vivado design in the form of a PYNQ overlay has been tested on a PYNQ-Z2 board under the Linux operating system. The neural network design runs at the 100MHz clock frequency. The synthesis and implementation provided valuable information on the consumption of FPGA resources and the feasibility of implementation. The results can be found at the end of the paper.