Updated

A senior administration official tells Fox News that Attorney General Holder asked the president Tuesday to invoke executive privilege and the president completely agreed that it was warranted in this case.

"The reason is the president believes the documents in question are "not anything material to [the] Fast and Furious [investigation]." Instead these documents are "internal deliberations within the Justice Dept" about how the administration planned to respond to congressional investigators and media inquiries about Fast and Furious - not the actual Fast and Furious investigation itself, per the same senior administration official.

"The president feels strongly this is a protected group of documents" because it falls under the principle of internal deliberations and advice within administration, noted the senior administration official. Presidents in both parties have asserted this same privilege.

A spokesman for House Speaker John Boehner, though, questioned whether Obama's assertion means White House officials may have been involved in Fast and Furious discussions.

"Until now, everyone believed that the decisions regarding 'Fast and Furious' were confined to the Department of Justice. The White House decision to invoke executive privilege implies that White House officials were either involved in the 'Fast and Furious' operation or the cover-up that followed. The Administration has always insisted that wasn't the case. Were they lying, or are they now bending the law to hide the truth?" Boehner spokesman Michael Steel said.

But the assertion does not necessarily mean that President Obama himself was involved in the internal deliberations. Remember that during the U.S. attorney scandal the Bush administration was asserting executive privilege for Karl Rove's conversations with Justice Department officials - this did not mean that then-President George W. Bush was involved in the conversations themselves.

Obama administration officials also point out that Bush asserted executive privilege six times and Clinton 14 times - both of whom protected the same category of documents being protected today. Dating back to President Reagan, presidents have asserted executive privilege 24 times.