A Muon Collider (MC) and a Neutrino Factory (NF) - which may be considered as a step towards a MC - both require a high-power (~4 MW) proton driver providing short (<1 m r.m.s. length) bunches for muon production. However, the driver repetition rates required for these two machines are different: ~15 Hz for MC and ~60 Hz for NF. This difference suggests employing two separate rings: one for accumulation of the proton beam from the Project-X linac in a few (e.g. 4) long bunches, the other for bunch compression - one by one for NF or all at a time for MC with simultaneous delivery to the target. The lat-tice requirements for these two rings are different: the momentum compaction factor in the accumulator ring should be large (and possibly negative) to avoid the mi-crowave instability, while the compressor ring can be nearly isochronous in order to limit the required RF volt-age and reduce the dispersion contribution to the beam size. In the present report we consider ring lattice designs which achieve these goals.