Americans also general believe the government has struck the right balance in its effort to fight terrorism. Just 36 percent say the government has gone too far in infringing on personal privacy, versus 46 percent who say the right balance has been struck and 13 percent who want additional government surveillance.

That said, nearly six in 10 Americans have some concern generally about their privacy being lost. But fewer than half describe themselves as "very" or "somewhat" concerned the government might be collecting their phone records or minoring their Internet use.

The poll also found that Americans are generally unconcerned that the public revelation of the NSA programs by 29-year-old defense contractor Edward Snowden would hurt anti-terror efforts. Six in 10 say the revelation did not weaken the United States's ability to prevent future attacks.

The complicated public opinion on the topic mirrors a fractured landscape in Washington, where strange pairings like Sen. Dianne FeinsteinDianne FeinsteinDem: Trump's China trademark looks like a quid pro quo Senate advances Trump's Commerce pick Flynn told FBI he didn't talk sanctions with Russian envoy: report MORE (D-Calif.) and Speaker John BoehnerJohn BoehnerFormer House leader Bob Michel, a person and politician for the ages Former House GOP leader Bob Michel dies at 93 Keystone pipeline builder signs lobbyist MORE (R-Ohio) have defended the NSA surveillance, while Sens. Rand PaulRand PaulTrump to interview four candidates for national security adviser Cruz backs former UN Ambassador John Bolton to replace Flynn GOP's ObamaCare talking points leave many questions unanswered MORE (R-Ky.) and Ron WydenRon WydenSenate Dems move to nix Trump's deportation order DNI confirmation hearing expected on Senate return Senate confirms Mnuchin as Treasury secretary MORE (D-Ore.) have expressed alarm.

It also provides new clarity to a Pew Poll, released Monday, that found 56 percent of Americans agreed the NSA phone record tracking was "an acceptable way for the government to investigate terrorism." Forty-one percent said it was unacceptable.

The CBS News poll was conducted June 9-19 among 1,015 adults, carrying a margin of error of 3 percentage points.