We consider the problem of finding a solution to the incompressible Euler equations \omega_{t} + v\cdot \nabla \omega = 0 \ \hbox{in }\mathbb{R}^{2} \times (0,\infty), \quad v(x,t) = \frac{1}{2\pi}\int_{\mathbb{R^2}}\frac{(y-x)^{\perp}}{|y-x|^{2}}\omega (y,t)\, dy that is close to a superposition of travelling vortices as t\to \infty . We employ a constructive approach by gluing classical travelling waves: at main order two vortex-antivortex pairs. Each pair consists of two vortices a distance 2q from each other, with wave speed c , moving in the positive and negative x_{1} directions respectively. More precisely, we find an initial condition that leads to a 4-vortex solution of the form \omega (x,t) = \omega_{0}(x-cte_{1}) - \omega_{0} ( x+ cte_{1}) + o(1) \quad \hbox{as }t\to\infty, where \omega_{0}( x ) = \frac{1}{\varepsilon^{2}}W \biggl ( \frac{x-q}{\varepsilon} \biggr ) - \frac{1}{\varepsilon^{2}}W \biggl ( \frac{x+q}{\varepsilon} \biggr ) + o(1) \quad \hbox{as }\varepsilon \to 0 and W(y) is a certain fixed smooth profile, radially symmetric, positive in the unit disk and zero outside.