  Let $n$ be a non-zero integer. A set $S$ of positive integers is a
Diophantine tuple with the property $D(n)$ if $ab+n$ is a perfect square for
each $a,b \in S$ with $a \neq b$. It is of special interest to estimate the
quantity $M_n$, the maximum size of a Diophantine tuple with the property
$D(n)$. In this notes, we show the contribution of intermediate elements is
$O(\log \log |n|)$, improving a result by Dujella. As a consequence, we deduce
that $M_n\leq (2+o(1))\log |n|$, improving the best-known upper bound on $M_n$
by Becker and Murty.
