Let $a$ be the smallest of these integers. Then $a + (a + 1)+(a + 2)\cdots + (a + 2007) = 251 \cdot (2a + 2007)\cdot 2^2$. In order for this to be a perfect square, we must have $2a+2007 =251n^2$ for some positive integer $n$. For $n = 1$ or $2$, $a$ is negative. For $n = 3$, we have $a = 126$ so that $a + 2007 = \boxed{2133}$ is the desired minimum value.