Update: William Branigin reports in the Washington Post:

President Obama said Friday he accepted the resignation of embattled Veterans Affairs Secretary Eric K. Shinseki.

He made the announcement shortly after Shinseki apologized publicly Friday for what he called an “indefensible” lack of integrity among some senior leaders of the VA health-care system and announced several remedial steps, including a process to remove top officials at the troubled VA medical center in Phoenix.

Veterans Affairs Secretary Eric Shinseki spoke Friday morning to the National Coalition for Homeless Veterans about what the VA is doing to prevent and treat homelessness in veterans, saying "now is not the time to let up or get complacent."

Speaking after a meeting with Shinseki at the White House, Obama said Shinseki had offered him his resignation.

“With considerable regret, I accepted,” Obama said. “We don’t have time for distractions,” he added. “We need to fix the problem.”

Veterans Administration Secretary Eric Shineski will meet with President Obama this morning for a "serious conversation" about his ability to remain in the administration.

The Hill:

The president said he would discuss whether Shinseki “has the capacity” to “take on the job of fixing” problems within the department in an interview taped Thursday with morning talk show hosts Kelly Ripa and Michael Strahan.

Shinseki is also expected to present the preliminary results of an internal review of the VA at the Oval Office meeting, which is closed to the press.

Speaking earlier Friday at an event for homeless veterans, Shinseki apologized for mismanagement within his department.

He said he would move to remove senior leaders at a Phoenix-area medical facility where veterans were kept from booking a primary care appointment an average of 115 days, and said bonuses would not be given to senior leaders in the department this year.

Shinskei also endorsed a Senate bill that would give him greater authority to fire top officials within his department.

But the question is whether Shinseki himself will survive the day. More than 100 members of Congress, including nearly a dozen Senate Democrats, have called for his ouster.

And on Thursday, White House press secretary Jay Carney pointedly refused to say that Shinseki had retained the president’s confidence.

Instead, Carney said Obama was seeking “accountability” after an initial inspector general report found systematic mismanagement of VA facilities.

“When it comes to the current situation, the inquiries and the investigations and some of the allegations, the president wants to see the results of these reports,” Carney said. “And he, as you know, made clear that he believes there ought to be accountability once we establish all the facts.”

Obama "will be very interested in the results," Carney added.

I think Shineski stays - for a while. I can't see President Obama appearing at a podium with Skineski after their meeting and tacitly accepting responsibility for the VA mess by announcing his resignation. Given the circumstances, it would also be humiliating for Shineski.

There's a possibility that Shineski would meet the press alone to say he was leaving, with Obama not deigning to make an appearance. But then why make a big deal about meeting with him in the first place?

The meeting is designed to take pressure off of Democrats on the Hill, not to determine if Shineski can continue. That is a political determination in which Shineski has no say. Obama hates to appear to be acting under pressure - the Sebelius resignation is a good example - so I think it probable that Shineski keeps his job for the time being.