In this work, a semianalytic solution for the acoustic streaming phenomenon, generated by standing waves in Maxwell fluids through a two-dimensional microchannel (resonator), is derived. The mathematical model is non-dimensionalized and several dimensionless parameters that characterize the phenomenon arise: the ratio between the oscillation amplitude of the resonator and the half-wavelength ($\eta =2A/\lambda _{a}$); the product of the fluid relaxation time times the angular frequency known as the Deborah number ($De=\lambda _{1}\omega$); the aspect ratio between the microchannel height and the wavelength ($\epsilon =2 H_{0}/\lambda _{a}$); and the ratio between half the height of the microchannel and the thickness of the viscous boundary layer ($\alpha =H_{0}/\delta _{\nu }$). In the limit when $\eta \ll 1$, we obtain the hydrodynamic behaviour of the system using a regular perturbation method. In the present work, we show that the acoustic streaming speed is proportional to $\alpha ^{2.65}De^{1.9}$, and the acoustic pressure varies as $\alpha ^{6/5}De^{1/2}$. Also, we have found that the growth of inner vortex is due to convective terms in the Maxwell rheological equation. Furthermore, the velocity antinodes show a high dependency on the Deborah number, highlighting the fluid's viscoelastic properties and the appearance of resonance points. Due to the limitations of perturbation methods, we will only analyse narrow microchannels.