But phone companies, the leading contenders to store the records, have said they don't want to that job, the current and former officials said.
The intelligence agency has come under enormous pressure for collecting and storing virtually all U.S. citizens' phone records as part of a growing domestic mass surveillance and collection program that came to light through documents leaked by former NSA contractor Edward Snowden.
Having third parties such as phone companies hold the data is one of the proposals the NSA is exploring in the hope that if it reforms itself, Congress won't halt the bulk-collection program altogether, the officials told the Post.
"The phone companies would run the analytics and provide you the analysis: 'Hey, this bad guy is talking to this bad guy,'" a former intelligence official told the newspaper.
Having phone companies analyze phone records on the government's behalf may still raise privacy, cost and other concerns, the Post said.
A presidential panel looking into both foreign and domestic U.S. spy practices last month called for custody of the phone records, known as metadata, to be moved to the phone companies or another third party.
The NSA would be required under the proposal to get a court order from a secret national security court to search the phone records.
President Obama said after receiving the advisory panel's 46 recommendations he was open to requiring phone companies to store the records and allowing the government to search them under strict guidelines.
Obama is reviewing the panel's recommendations and has promised to lay out his own set of intelligence and surveillance reforms this month. The Post said Wednesday he could do that as early as next week.
NSA Director Gen. Keith Alexander told the presidential panel the NSA was "seriously" considering "moving to a model in which the data are held by the private sector," a review group member told the newspaper.
But Alexander said a problem was, "No one else wanted it -- especially not the phone companies," the group member said.
Alexander described the matter as a "bit of a hot potato," the person said.
The phone companies say they don't want to be made to hold the data for the NSA longer than they normally would, the Post said.
They also say the stored records would likely become an attractive target for hackers.
One carrier said it would probably cost $50 million a year to maintain a five-year, searchable database, a company official told the Post two weeks ago.
Key senators who studied the private-sector idea -- including the option of paying the companies to keep the data -- have rejected it, the Post said.
Civil libertarians consider mandated phone-company or third-party storage an unacceptable "proxy" for the NSA's holding of the database.
White House National Security Council spokeswoman Caitlin Hayden said Obama would make his remarks about NSA reforms before he delivers his State of the Union address to Congress Jan. 28.
As of now, the White House is "continuing to study" the panel's report and gather ideas from the intelligence agencies and industry, she told the Post.