The following is an attempt to validate the conclusion proposed by
@vidyarthi.

Theorem: Every
doubly-stochastic partial-isometric matrix is the product of a
permutation matrix and a doubly-stochastic projection.

Proof:
Given a doubly-stochastic partial-isometric matrix $A$, one has that $A^tA$ and $AA^t$ are doubly-stochastic
projections, so by Theorem 2 in (*Sinkhorn, R.*, **Two results concerning doubly stochastic matrices**, Am. Math. Mon. 75, 632-634 (1968). ZBL0162.04205) there are permutation matrices $U$ and $V$ such that
$$
U^tA^tAU = P(k_1)\oplus P(k_2)\oplus \cdots \oplus P(k_n)
$$
and
$$
V^tAA^tV = P(l_1)\oplus P(l_2)\oplus \cdots \oplus P(l_m),
$$
where, for any integer $k$,
$$
P(k):=
\pmatrix{
1/k & 1/k & \ldots & 1/k \cr
\vdots & \vdots & \ddots & \vdots\cr
1/k & 1/k & \ldots & 1/k \cr}.
$$
Replacing $A$ with $V^tAU$, we may assume that $U$ and $V$ coincide with the identity matrix and hence $U$ and $V$ will henceforth
be ommitted.

Set $c(k)=(1/\sqrt k,1/\sqrt k,…,1/\sqrt k) ∈ \mathbb R^k$, so that $c(k)$ is a unit vector spanning the range of $P(k)$.
Moreover the range of the
projection $A^tA$ above admits an orthonormal basis formed by the vectors
$$
u_1 = c(k_1)\oplus 0_{k_2}\oplus \cdots \oplus 0_{k_n},
$$
$$
u_2 = 0_{k_1}\oplus c(k_2)\oplus \cdots \oplus 0_{k_n},
$$
$$...$$
$$
u_n = 0_{k_1}\oplus 0_{k_2}\oplus \cdots \oplus c(k_n),
$$
a similar reasoning yielding a basis $\{v_1, v_2, …, v_m\}$ for the range of $AA^t$. The initial and final
projections of a partial isometry share rank, so $n=m$, and we claim that, up to a permutation of indices $i$, one has
that $k_i=l_i$, for all $i$.

Notice that $A$ maps $\hbox{span}\{u_i\}$ isometrically onto $\hbox{span}\{v_i\}$ so, for $i\neq j$, one has that
$Au_i$ and $Au_j$ are orthogonal vectors. However these vectors have nonnegative coordinates so their support (set of
indices for nonzero coordinates) must be disjoint. By the pigeonhole principle each $u_i$ must therefore be mapped
under $A$ to a scalar
multiple of some $v_j$. By positivity and norm preservation these scalars must coincide with 1 so there is a permutation $\sigma $ such that $Au_i=v_{\sigma (i)}$, for all $i$.

Observe that, being doubly-stochastic, $A$ leaves invariant the linear functional $\Sigma$ which sums all of the coordinates of a vector. Noticing that $\Sigma(u_i)=\sqrt{k_i}$, while $\Sigma(v_j)=\sqrt{l_j}$, we deduce that $k_i=l_{\sigma (i)}$.

It is now easy to see that there exists a permutation matrix $W$ such that $Wu_i = v_{\sigma (i)}$. Letting
$$
B=W^tA,
$$
we then have that $Bu_i=u_i$, while $B^tB=A^tA$.

It follows that $B$ is a partial isometry coinciding with the identity operator on its initial space, and hence that $B$ coincides with its initial projection $B^tB$. This leads to
$$
A=WB=WB^tB=WA^tA.
$$