WASHINGTON—The Trump administration and lawmakers in both parties have failed so far to find a path forward to renew a set of key domestic surveillance powers due to lapse in 11 days, prompting heightened alarm among intelligence agencies that they will soon lose tools they consider vital to national security.

President Trump met with Attorney General William Barr and senior Republicans on Tuesday in an effort to hammer out a deal to address privacy concerns raised by Mr. Trump—who remains convinced spying tools were improperly...