  Let $r\geq 1$ be a positive integer, $A$ a real positive definite symmetric
$r\times r$ matrix, $B$ a vector of length $r$, and $C$ a scalar. Nahm's
problem is to describe all such $A,B$ and $C$ with rational entries for which a
specific $r$-fold $q$-hypergeometric series (denoted by $f_{A,B,C}(q)$)
involving the parameters $A,B,C$ is modular. When the rank $r=2$, Zagier
provided eleven sets of examples of $(A,B,C)$ for which $f_{A,B,C}(q)$ is
likely to be modular. We present a number of Rogers--Ramanujan type identities
involving double sums, which give modular representations for Zagier's rank two
examples. Together with several known cases in the literature, we verified ten
of Zagier's examples and give conjectural identities for the remaining example.
