The poll results reflect a deep sense of skepticism among Americans about the security of their information, said Ipsos pollster Chris Jackson.
Privacy concerns have grown in response to revelations about U.S. government surveillance programs as well as a constant stream of high-profile security breaches that compromised consumer records including credit cards numbers, email logins and medical information, he said.
"People are very distrusting of everybody, but Americans actually trust Apple a bit more than the government on some issues," Jackson said.
The Reuters/Ipsos poll shows that large numbers of Americans want to keep their phone records, text messages, emails and other Internet activity private.
For example in this month's poll, 69 percent said they would not give up email privacy even if it would help the government foil foreign terror plots and 75 percent said they would be unwilling to give up text-message privacy for the same reason.
Opinion on whether Apple is right is divided by political party lines: 54 percent of Democrats agree with Apple, while only 37 percent of Republicans support the company.
Donald Trump, front-runner for the Republican presidential nomination, last week said he would boycott the company's products until it unlocks the phone.
Democratic U.S. Representative Ted Lieu on Tuesday asked the Federal Bureau of Investigation to rescind the unlock order.
There is this tension: Americans want terrorists to be prosecuted, but in the context of issues about security and privacy, it becomes a much more nuanced discussion," Jackson said.