WASHINGTON—President Barack Obama, after a day of withering criticism over the government’s handling of the Ebola virus, said Thursday he may name a point person to oversee the administration’s response and is open to a travel ban but isn’t planning one.

His evening remarks, which sought to show a more forceful response while assuring Americans they remain safe, were the first sign the White House could adopt ideas that Republican lawmakers have emphasized in ratcheting up their attacks this week. They have called for creating...