Obama said the companies needed to do more if they were to qualify for further government bailout funds.
"They're not there yet," Obama told CBS television in an interview aired on Sunday.
"We think we can have a successful US auto industry. But it's got to be one that's realistically designed to weather this storm and to emerge - at the other end - much more lean, mean and competitive than it currently is."
Last week, Obama cited mismanagement "over the years" for some of the car industry's financial problems.
The industry has been pushed to the brink of bankruptcy, attributed to the global economic crisis, as car sales decline.
GM and Chrysler have asked for another $21.6bn on top of the $17.4bn in emergency loans approved in December.
Wagoner, who was at the helm of the largest US carmaker for eight years and who had said last week that he had no plans to quit, was asked by the White House to go, the US media said.
A statement from GM said that Fritz Henderson, the compnay's chief operating officer and the architect of the carmaker's existing turnaround plan, will take over as CEO.
Wagoner had come under fire for his leadership of ailing company late last year when US politicians debated a bailout for the carmaker.
Ford, the other member of so-called "Big Three" US carmakers, has said it has enough money to survive the downturn without government aid.