  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.
