  Let $I$ be a perfect ideal of height two in $R=k[x_1, \ldots, x_d]$ and let
$\varphi$ denote its Hilbert-Burch matrix. When $\varphi$ has linear entries,
the algebraic structure of the Rees algebra $\mathcal{R}(I)$ is well-understood
under the additional assumption that the minimal number of generators of $I$ is
bounded locally up to codimension $d-1$. In the first part of this article, we
determine the defining ideal of $\mathcal{R}(I)$ under the weaker assumption
that such condition holds only up to codimension $d-2$, generalizing previous
work of P.~H.~L.~Nguyen. In the second part, we use generic Bourbaki ideals to
extend our findings to Rees algebras of linearly presented modules of
projective dimension one.
