RICHMOND, Va. — The besieged Democratic governor of Virginia, Ralph Northam, began to emerge from his defensive crouch on Tuesday, and signaled through his diminished corps of allies that he believed he could remain in office and, perhaps, prove that he did not appear in a racist photograph more than 30 years ago.

The governor, whose power has been on the verge of collapse since the photograph surfaced on Friday, intends to hire a private investigator to examine the circumstances of the picture, according to a Democrat familiar with his plans. The image appeared on Mr. Northam’s medical school yearbook page and showed a person dressed in blackface and another in a Ku Klux Klan robe.

[Read about the controversial history of the Northam yearbook.]

And Mr. Northam’s supporters have begun to express increased confidence that he will hold on to his job, despite a national onslaught of condemnation, in part because the Republicans who control the General Assembly have demonstrated little urgency toward seeking his ouster.

“No matter the veracity of the photo in question, a man who has devoted his entire life to the service of others should have the opportunity to clear his name,” State Senator Richard H. Stuart, a Republican who is close to Mr. Northam, said in a statement of support. “People who have been elected to represent Virginians should have the courage to say what is self-evident to so many people — that poor judgment 34 years ago should not outweigh a selfless service to people from every walk of life.”