The High Luminosity LHC (HL-LHC) will require upgrades to the ATLAS Level-0 (L0) muon trigger to cope with the increased collision rates. The Level-0 muon trigger will incorporate high precision hits from the Monitored Drift Tube chambers (MDTs) for the first time. The MDT Trigger Processor (MDTTP) uses the MDT hits to reconstruct muon tracks. This results in a reduction of the L0 trigger rate due to a sharpened efficiency turn-on curve, plus a rejection of combinatoric fakes. All MDT front-end electronics will be replaced during the HL-LHC upgrades necessitating new back-end electronics. The MDTTP will be used as the new MDT back-end. The MDTTP is an ATCA blade based on the Apollo platform. The 64 MDTTP blades also configure and monitor the MDT front-end electronics and read out MDT data for the final acquisition. An overview of the trigger algorithm and performance along with the status of the MDTTP hardware and firmware development and testing will be presented.