FBI director James Comey notified Congress Friday that investigators were examining whether newly discovered emails may be relevant to the dormant investigation of Clinton's use of private emails while secretary of state.
Trump said emails - the contents of which are unknown - will be "absolutely devastating" to Clinton.
Comey has said he released more information about the investigation than is typical because of election year sensitivities.
The officials were not authorized to discuss the matter by name and spoke on condition of anonymity.
Earnest said he wouldn't judge Comey, appointed by Obama in 2013, for making the announcement 11 days before Election Day, which has earned near universal criticism from fellow Democrats.
Meanwhile, the presidential race has tightened dramatically in the last several days.
"Thank you, Huma. Good job, Huma", Trump said, referring to Human Abedin, a top Clinton aide. The FBI is still working to determine if any of them are pertinent to the investigation into Clinton's private email server.
And his announcement means that Clinton could have to contend with the news that the FBI has resumed its investigation of her use of a private email server - without any clarity on whether its investigators will find anything significant - up to and beyond Election Day. Comey concluded in July that while Clinton and her staff had been "extremely careless" in handling classified information, there were no grounds for any charges.
Justice Department officials who were advised of the FBI's intention to notify Congress about the discovery expressed concern that the action would be inconsistent with department protocols created to avoid the appearance of interference in an election.
Earnest said the White House did not get advance notice of Comey's letter being sent to Congress on Friday.
The move creates the potential for a divide between the Justice Department and Comey, who has served in government under both Democratic and Republican presidents.
It was not immediately clear what the emails were about or what significance, if any, they carried to the email investigation. Nor was it clear when agents would be able to fully review the recovered emails, and Comey made no guarantees that would happen before Election Day.
The emails were found on a laptop of Abedin's estranged husband, Anthony Weiner, who is now under investigation for allegedly sexting an underage girl. As part of the process in 2015 of returning her work-related emails to the State Department, Abedin said she looked for all the devices that may have any of my State Department work and provided two laptops and a Blackberry to her lawyers for review. The person said it is "news to (Abedin)" that her emails would be on a computer belonging to her husband.
If the Federal Bureau of Investigation finds emails Abedin sent or received through the clintonemail.com server archived on the device recently recovered from her home, that would appear to conflict with what she told the Federal Bureau of Investigation earlier this year.
The answer is not yet clear, but its possible that either she did not know about the emails on the computer of her estranged husband, forgot about them or for some other reason did not turn them over. That was my practice for all my email accounts. They turned materials in, and I know they did so. I had a few folders, but they were not deleted.