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The newspaper reported on its website late on Sunday that Facebook is looking at Gibbs, once one of President Barack Obama's closest aides, to help manage communications in advance of an initial public offering planned for early next year. The Times cited people briefed on the talks.
The sources told the Times the talks were at an early stage and there was no formal offer yet.
The Times quoted the sources as saying that Gibbs, who left his White House job last month after a two-year stint, had planned to help establish Obama's 2012 re-election campaign before moving to the private sector.
But Facebook wants Gibbs to consider the matter more quickly, said the sources, who requested anonymity.
The Times said both Facebook and Gibbs declined comment.