2D inverse Fourier transform of $\frac{\mathrm{exp}(-|\mathbf{k}|^{2})}{1+|\mathbf{k}|^{2}}$

Question

I enconter a problem in 2D fluid mechanics, where I need the 2D inverse Fourier transform of the following function: $$\frac{\mathrm{exp}(-|\mathbf{k}|^{2})}{1+|\mathbf{k}|^{2}},$$ where $\mathbf{k}=(k_{x}, k_{y})$, $|\mathbf{k}|^{2}=k_{x}^{2}+k_{y}^{2}$. The inverse Fourier transform of a function is defined as: $$ F(x, y)=\int_{-\infty}^{\infty} \int_{-\infty}^{\infty} f\left(k_x, k_y\right) e^{-i\left(k_x x+k_y y\right)} d k_x d k_y. $$ I searched around it seems related to the Bessel function and error function but as it is 2D, I am really not familiar with it, can anyone help?