In this work, we theoretically study the interaction between a solid state membrane equipped with a nanopore and a tethered, negatively charged polymer chain subjected to a time-dependent applied electrolyte bias. In order to describe the movement of the chain in the biomolecule-membrane system immersed in an electrolyte solution, Brownian dynamics is used. We show that we can control the polymer's equilibrium position with various applied electrolyte biases: for a sufficiently positive bias, the chain extends inside the pore, and the removal of the bias causes the polymer to leave the pore. Corresponding to a driven process, we find that the time it takes for a biomolecular chain to enter and extend into a nanopore in a positive bias almost increases linearly with chain length while the amount of time it takes for a polymer chain to escape the nanopore is mainly governed by diffusion.