We show that if T is an $varepsilon$-approximate Jordan functional such that T(a) = 0 implies $T(a^2) = 0 (a in A)$ then T is continuous and $Vert T Vert leq 1 + varepsilon$. Also we prove that every $varepsilon$-near Jordan mapping is an $g(varepsilon)$-approximate Jordan mapping where $g(varepsilon) o 0$ as $varepsilon o 0$ and for every $varepsilon > 0$ there is an integer m such that if T is an $frac {varepsilon}{m}$-approximate Jordan mapping on a finite dimensional Banach algebra then T is an $varepsilon$-near Jordan mapping.