Polymer dynamics in a turbulent flow is a problem spanning several orders of magnitude in length and time scales. A microscopic simulation covering all those scales from the polymer segment to the inertial scale of turbulence remains improbable within the foreseeable future. We propose a multiscale simulation strategy to enhance the spatio-temporal resolution of the local Lagrangian turbulent flow by matching two different simulation techniques, i.e. direct numerical simulation for the flow as a whole, and the lattice Boltzmann method coupled to polymer dynamics at the Kolmogorov dissipation scale. Local turbulent flows sampled by Lagrangian tracer particles in the direct numerical simulation are reproduced in the lattice Boltzmann model with a finer resolution, by supplying the latter with both the correct initial condition as well as the correct time-dependent boundary condition, sampled from the former. When combined with a Molecular Dynamics simulation of a polymer chain in the lattice Boltzmann model, it provides a strategy to simulate the passive dynamics of a polymer chain in a turbulent flow covering all these scales. Although this approach allows for a fairly realistic model of the macromolecule, the back-coupling to the flow on the large scales is missing.