Highly selective first-level triggers are essential to exploit the full physics potential of the ATLAS experiment at High-Luminosity LHC (HL-LHC). The concept for a new muon trigger stage using the precision monitored drift tube (MDT) chambers to significantly improve the selectivity of the first-level muon trigger is presented. It is based on fast track reconstruction in all three layers of the existing MDT chambers, made possible by an extension of the first-level trigger latency to six microseconds and a new MDT read-out electronics required for the higher overall trigger rates at the HL-LHC. Data from $pp$-collisions at $sqrt{s} = 8,mathrm{TeV}$ is used to study the minimal muon transverse momentum resolution that can be obtained using the MDT precision chambers, and to estimate the resolution and efficiency of the MDT-based trigger. A resolution of better than $4.1%$ is found in all sectors under study. With this resolution, a first-level trigger with a threshold of $18,mathrm{GeV}$ becomes fully efficient for muons with a transverse momentum above $24,mathrm{GeV}$ in the barrel, and above $20,mathrm{GeV}$ in the end-cap region.
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