The Federal Trade Commission (FTC) has been investigating allegations that political consultancy Cambridge Analytica improperly obtained the data of up to 87 million Facebook users.
The settlement was approved by the FTC in a 3-2 vote, sources told US media.
Facebook and the FTC told the BBC they had no comment on the reports.
How was the settlement reached?
The consumer protection agency the FTC began investigating Facebook in March 2018 following reports that Cambridge Analytica had accessed the data of tens of millions of its users.
The investigation focused on whether Facebook had violated a 2011 agreement under which it was required to clearly notify users and gain "express consent" to share their data.
The $5bn fine was approved by the FTC in a 3-2 vote which broke along party lines, with Republican commissioners in favour and Democrats opposed.
The New York Times reported that the Democrats wanted stricter limits on the firm, while other Democrats have criticised the fine as inadequate.
"With the FTC either unable or unwilling to put in place reasonable guardrails to ensure that user privacy and data are protected, it's time for Congress to act," US Senator Mark Warner said.
The fine still needs to be finalised by the Justice Department's civil division, and it is unclear how long this may take, the sources said.
If confirmed, it would be the largest fine ever levied by the FTC on a tech company.
However, the amount falls in line with estimates by Facebook, which earlier this year said it was expecting a fine of up to $5bn.
Investors responded positively to the news, pushing Facebook shares up 1.8%.