His resignation is the latest example of the government taking a hands-on role in making major decisions at companies it is bailing out. The government has already pushed banks to make management changes and sharply reduce or eliminate their dividends, and it also is directing many of the decisions at the troubled insurance giant American International Group, which is nearly 80 percent owned by the government after its rescue.

In deciding to urge Mr. Wagoner to step down, the Obama administration seemed mindful of the public’s growing outrage over bailouts of private companies, as well as the bonuses paid to employees of A.I.G.

Mr. Obama is well aware that he cannot afford to give the appearance of using tax dollars to reward executives who have done a poor job, and he began signaling as early as last week that he would take a tough stance with the automakers.

Image Robert Nardelli, the chief of Chrysler, at the North American International Auto Show in Detroit in January. Credit... Mark Blinch/Reuters

In a question and answer session at the White House on Thursday, the president said there had been “a lot of mismanagement of the auto industry over the past several years,” and declared that more government help would be contingent on the companies’ “willingness to make some pretty drastic changes.”

The plan Mr. Obama is to announce on Monday will also include government backing of warranties for G.M. and Chrysler cars and trucks, to give consumers enough confidence to buy them, even if one or both are forced into bankruptcy.

In Detroit, the G.M. board said Monday in a statement that it “has recognized for some time that the company’s restructuring will likely cause a significant change in the stockholders of the company and create the need for new directors with additional skills and experience.”