The White House has begun a quiet self-assessment in the wake of the troubled health-law launch, recognizing that administration officials missed warning signs and put too much trust in their management practices in implementing a program that is the centerpiece of President Barack Obama’s domestic legacy…

Some of Mr. Obama’s allies are concerned that he wasn’t aware of the details of several matters that have become problematic for his administration. Mr. Obama said he didn’t know of the problems with the health-care website, and administration officials have said he was caught off guard when leaked security files revealed the U.S. was gathering information from German Chancellor Angela Merkel’s cellphone.

It is unclear whether Mr. Obama will shake up his senior staff of longtime and trusted aides. David Axelrod, a former senior White House adviser, said the president “is someone who maintains an equilibrium. He’s not a guy who loses it.…I think at the end of the day, he’s always taken the long view, so his interest is in fixing this and seeing this thing through.”