One has to be careful with perturbations of metrics of nonnegative curvature, because that may introduce negative curvature.

Here's another approach which gives you a nonnegatively curved metric.

Start with $S^3\times S^2$ with the product of round metrics.  (Note that the round metric on S^3 is its biinvariant metric).

Consider the $S^1$ action on $S^3 \times S^2$ where it acts as the Hopf action on $S^3$ and simultaneously rotates the $S^2$ factor $2k$ times around for some integer $k$.

To make it explicit, thinking of $S^2$ as the unit sphere in the imaginary quaternions, the action can be described as $z*(p,q) = (zp, z^k q \overline{z}^k)$.

The action is clearly free and the quotient is diffeomorphic to $S^2 \times S^2$.  Since the circle is acting isometrically, there is an induced submersion metric on $S^2\times S^2$.  By the O'Neill formulas for a submersion, this metric has nonnegative curvature.  When k = 0, one gets the usual product of round metrics, but when $k\neq 0$ the metric is, in general, not a product.