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 Emiel FeinsteinSenate left in limbo by Trump tweets, House delays Children should not be human shields against immigration enforcement The Hill's Morning Report — Sponsored by PhRMA — Immigration drama grips Washington MORE (D-Calif.) and Speaker John BoehnerJohn Andrew BoehnerFormer top Treasury official to head private equity group GOP strategist Steve Schmidt denounces party, will vote for Democrats Zeal, this time from the center MORE (R-Ohio) have defended the NSA surveillance, while Sens. Rand PaulRandal (Rand) Howard PaulGOP senators call for probe of federal grants on climate change Overnight Health Care — Presented by the Association of American Medical Colleges — Key ObamaCare groups in limbo | Opioids sending thousands of kids into foster care | House passes bill allowing Medicaid to pay for opioid treatments US watchdog: 'We failed' to stem Afghan opium production MORE (R-Ky.) and Ron WydenRonald (Ron) Lee WydenScrutiny ramps up over Commerce secretary's stock moves Hillicon Valley: Justices require warrants for cellphone location data | Amazon employees protest facial recognition tech sales | Uber driver in fatal crash was streaming Hulu | SpaceX gets contract to launch spy satellite On The Money — Sponsored by Prudential — Supreme Court allows states to collect sales taxes from online retailers | Judge finds consumer bureau structure unconstitutional | Banks clear Fed stress tests 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.