Gibbs left the White House in February after two years on the job, ostensibly to work on President Obama's re-election campaign, The New York Times reported.
Neither Gibbs nor Facebook would comment on the situation and the Times said its sources spoke on condition of anonymity.
Sources told the Times Facebook was pressing Gibbs to consider the job ahead of Facebook's expected initial public offering next year. The position could be worth millions of dollars although details of a compensation package had not been discussed.
Some investors have valued Facebook at more than $60 billion.
Gibbs has reportedly discussed the offer with several former White House colleagues, including David Axelrod, Obama's former senior adviser. Axelrod is now a member of Obama's re-election team.
The attention Facebook received from the movie "The Social Network" and investor interest has put additional stress on the California company to better communicate with the public, the Times said.
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