Three, if there exist 2007 real numbers $a_{1}$, $a_{2}, \cdots, a_{2007}$ that satisfy the given conditions, let $\left|\sum_{i=1}^{2007} a_{i}\right|=t \geqslant 0$.
Removing the absolute value symbol and separating the positive and negative parts, we can write
$$\left|\sum_{i=1}^{2007} a_{i}\right|=\sum_{i=1}^{k} x_{i}-\sum_{j=1}^{2007-k} y_{j}=t,$$

where $x_{1}, x_{2}, \cdots, x_{k}, y_{1}, y_{2}, \cdots, y_{2007-k}$ is a permutation of $\left|a_{1}\right|,\left|a_{2}\right|, \cdots,\left|a_{2007}\right|$.
Thus, by condition (2), we get
$\sum_{i=1}^{k} x_{i}+\sum_{j=1}^{2007-k} y_{j}=2006+t$.
Hence, $\sum_{i=1}^{k} x_{i}=1003+t, \sum_{j=1}^{2007-k} y_{j}=1003$.
Since $0 \leqslant x_{i}\sum_{i=1}^{k} x_{i}=1003+t$,
$2007-k>\sum_{j=1}^{2007-k} y_{j}=1003$.
This implies, $k \geqslant 1004,2007-k \geqslant 1004$.
Adding these, we get $2007 \geqslant 2008$, a contradiction.
Therefore, 2007 real numbers that satisfy the conditions do not exist.