<p>In this paper, a special class of 2D MHD equations with no velocity dissipation and no magnetic diffusion is studied, where the velocity equation contains the vertical damping and the magnetic field possesses the damping. In the absence of a magnetic field, the corresponding vorticity equation is a 2D Euler equation with an additional Riesz transformation term. The global well-posedness and the stability for the Euler-like equation remain an extremely challenging open problem. An essential difficulty lies in the control of the growth of Navier–Stokes nonlinear term caused by the lack of dissipation. In the paper, attention focuses on the coupling of Euler-like equation and magnetic equation. To solve the derivative-loss problem, we consider the perturbation of the MHD system near a background magnetic field. By making full use of smooth and stabilizing effect of the background magnetic field, we obtain the stability of the solution in Sobolev-setting <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(H^3(\mathbb {R}^2)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>H</mi> <mn>3</mn> </msup> <mrow> <mo stretchy="false">(</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mn>2</mn> </msup> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. In addition, the optimal decay rates in <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(H^2(\mathbb {R}^2)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>H</mi> <mn>2</mn> </msup> <mrow> <mo stretchy="false">(</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mn>2</mn> </msup> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> are also established.</p>