WASHINGTON—The U.S. Senate on Saturday defeated two attempts by Democrats to extend the Bush-era tax cuts for the middle class permanently. After the Senate votes, President Barack Obama told Democratic congressional leaders he would be open to a temporary extension of the Bush-era tax cuts for the affluent, but he would demand concessions from the GOP.

In rare weekend votes that likely had little effect on wider negotiations to reach a compromise about extending the tax cuts, the Senate voted 53-36 to reject an attempt to...