The Nevada Democrat was citing comments made during Tuesday's caucus luncheon by New Mexico Democrat Martin Heinrich.
Asked about a bipartisan, bicameral proposal from a pair of Texas lawmakers, Senate Minority Whip John Cornyn and Democratic Rep. Henry Cuellar, Reid said that based on what he had seen, "the answer from me is no, I won't support it."
"Our number one concern should be this narrow issue of how we take care of this situation we have on the border," Reid said. "As I've been told, the Cornyn-Cuellar legislation covers a lot of other issues ... Remember, had we done comprehensive immigration reform, we wouldn't be having this issue."
Cornyn and Cuellar jointly briefed reporters on their plan Monday evening. The House may roll out a border funding proposal in response to President Barack Obama's $3.7 billion request by the end of the week.
Asked if the Senate might depart for the August recess before moving the supplemental spending package, Reid said "I hope not."