The expanded program is designed to double the number of children covered, from 4 million to 8 million.
Eighteen Republicans joined all of the Democrats in voting to expand the program from its current annual budget of $5 billion to $12 billion for the next five years.
The reauthorization bill "fails to focus on poor children, and instead creates a new entitlement program for higher-income households," said White House Press Secretary Dana Perino in a written statement.
"The president will veto this bill because it directs scarce funding to higher incomes at the expense of poor families."
With the current program scheduled to expire Saturday, the White House encouraged Congress to send the president a continuing resolution extending the program.
"We should take this time to arrive at a more rational, bipartisan SCHIP reauthorization bill that focuses on children in poor families who don't currently have insurance, rather than raising taxes to cover people who already have private insurance," Perino added.
Bush and many Republicans contend the program's original intent would be changed under the current bill. The program is supposed to give parents who make too much to qualify for Medicaid but not enough to buy private insurance coverage for their children. The concern is that parents might be prompted to drop private coverage their children already have in order to get cheaper coverage under the bill.