We present a theoretical study of the quantum transport through a nanoscale system in which a central quantum dot (QD) is coupled asymmetrically to normal leads and to two Majorana bound states (MBSs) localized at the ends of a topological superconducting nanowire threaded by a tunable magnetic flux. The effects of the leads–QD coupling asymmetry parameter α and the bias voltage asymmetry parameter q on the system's linear conductance are considered for the case of unhybridized and hybridized MBSs. In the zero-temperature limit, for unhybridized MBSs the system's linear conductance is finite only when the magnetic flux phase φ = (2n+1)π (n∈Z) and it scales as G=2qα e²/[h(α+1)], while for hybridized MBSs it presents a complicated dependence on the system's parameters. At finite temperature, for unhybridized MBSs, the system's linear conductance oscillates as a function of the magnetic flux phase φ with a period of 2π, and the position of the linear conductance maxima can be shifted from φ=2nπ to φ=(2n+1)π by simply varying the value of the bias voltage asymmetry parameter q. For hybridized MBSs, the conductance exhibits a similar behavior when the energy level of the central QD, ε_d, is tuned at the leads' Fermi level (ε_d=ε_F), although when ε_d≠ε_F the oscillation period changes to 4π, and the position of the linear conductance maxima depends on the actual value of ε_d and other parameters in the system. Our results highlight the experimental importance of the leads-QD and bias voltage asymmetry parameters, which are often present in realistic experimental setups, and can strongly affect the identification and observation of MBSs transport signatures.