By Janet Hook and Sarah Portlock

Some senior Republicans in Congress on Sunday threw cold water on a cornerstone of President Barack Obama’s plan to revamp the National Security Agency’s surveillance programs–his proposal to provide a new advocate for privacy concerns.

House Homeland Security Committee Chairman Michael McCaul (R., Texas) raised questions about the proposal, which would add an advocate in the proceedings of the secret court that oversees the agency’s sweeping phone-data collection. He said he feared it would slow down antiterrorism efforts when time is of the essence.

Mr. McCaul, who dealt with these issues as a counterterrorism prosecutor, said changes like those Mr. Obama proposed would “slow down the efficacy and efficiency of our counterterrorism investigation.”