The notion of strong uniform convergence on bornologies introduced in 2009 by Beer-Levi turns to give the classical convergence introduced by Arzel\`a in 1883. Evert in 2003 introduced the notion of Arzel\`a-Whitney or simply AW-convergence for a net of functions. We define a new type of convergence, a "strong" form of Whitney convergence on bornologies, and we prove that on some families it coincides with that AW-convergence. Furthermore, we study the countability properties of this new function space.