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