The åisebox-0.5ex~39-m European Extremely Large Telescope (E-ELT) will be the largest telescope ever built. This makes it particularly suitable for sensitive polarimetric observations, as polarimetry is a photon-starved technique. However, the telescope mirrors may severely limit the polarimetric accuracy of instruments on the Nasmyth platforms by creating instrumental polarisation and/or modifying the polarisation signal of the object. In this paper we characterise the polarisation effects of the two currently considered designs for the E-ELT Nasmyth ports as well as the effect of ageing of the mirrors. By means of the Mueller matrix formalism, we compute the response matrices of each mirror arrangement for a range of zenith angles and wavelengths. We then present two techniques to correct for these effects that require the addition of a modulating device at the ``polarisation-free’’ intermediate focus that acts either as a switch or as a part of a two-stage modulator. We find that the values of instrumental polarisation, Stokes transmission reduction and cross-talk vary significantly with wavelength, and with pointing, for the lateral Nasmyth case, often exceeding the accuracy requirements for proposed polarimetric instruments. Realistic ageing effects of the mirrors after perfect calibration of these effects may cause polarimetric errors beyond the requirements. We show that the modulation approach with a polarimetric element located in the intermediate focus reduces the instrumental polarisation effects down to tolerable values, or even removes them altogether. The E-ELT will be suitable for sensitive and accurate polarimetry, provided frequent calibrations are carried out, or a dedicated polarimetric element is installed at the intermediate focus.