In a statement provided to CBS News, White House Counsel Gregory Craig says Pres. Obama is "very sympathetic to those who want to find out what happened."
But at the same time, Craig makes it clear that Mr. Obama is not disputing the claim of privilege.
"He is also mindful as president of the United States not to do anything that would undermine or weaken the institution of the presidency," Craig says in the statement.
A White House spokesman says the Counsel's Office is still studying the question of executive privilege.
Rove is under subpoena to testify to the House Judiciary Committee on February 23 about political motivations in the firings of U.S. Attorneys.
During the last days of the Bush presidency, White House Counsel Fred Fielding informed Rove's lawyer, Robert Luskin, that Mr. Bush was continuing to assert "executive privilege" and directed that Rove not testify or provide documents about the U.S. Attorney firings – even after Pres. Bush left office.
After Barack Obama was inaugurated, Luskin asked the new White House Counsel to advise on "whether there is a valid claim of executive privilege."
Craig's carefully worded statement straddles the fence on that point. His statement says Pres. Obama backs the investigation of the U.S. Attorney firings – but doesn't challenge whether "executive privilege" can be claimed after a President leaves office.
Craig says Pres. Obama "is urging" Rove and the House Judiciary Committee to settle the matter of privilege and testimony on their own.