  We develop a method for approximating the Gr\"obner basis of the ideal of
polynomials which vanish at a finite set of points, when the coordinates of the
points are known with only limited precision. The method consists of a
preprocessing phase of the input points to mitigate the effects of the input
data uncertainty, and of a new "numerical" version of the Buchberger-M\"oller
algorithm to compute an approximation $\bar{GB}$ to the exact Gr\"obner basis.
This second part is based on a threshold-dependent procedure for analyzing from
a numerical point of view the membership of a perturbed vector to a perturbed
subspace. With a suitable choice of the threshold, the set $\bar{GB}$ turns out
to be a good approximation to a "possible" exact Gr\"obner basis or to a basis
which is an "attractor" of the exact one. In addition, the polynomials of
$\bar{GB}$ are "sufficiently near" to the polynomials of the extended basis,
introduced by Stetter, but they present the advantage that $LT(\bar{GB})$
coincides with the leading terms of a "possible" exact case. The set of the
preprocessed points, approximation to the unknown exact points, is a pseudozero
set for the polynomials of $\bar{GB}$.
