A concept of multilevel thermally assisted magnetoresistive random-access memory is proposed and investigated by micromagnetic simulations. The storage cells are magnetic tunnel junctions in which the storage layer is exchange biased and in a vortex configuration. The reference layer is an unpinned soft magnetic layer. The stored information is encoded via the position of the vortex core in the storage layer. This position can be varied along two degrees of freedom: the radius and the in-plane angle. The information is read out from the amplitude and phase of the tunnel magnetoresistance signal obtained by applying a rotating field on the cell without heating the cell. Various configurations are compared in which the soft reference layer consists of either a simple ferromagnetic layer or a synthetic antiferromagnetic sandwich (SAF). Among those, the most practical one comprises a SAF reference layer in which the magnetostatic interaction between the SAF and storage layer is minimized. This type of cell should allow one to store at least 40 different states per cell representing more than five bits per cell.