Joseph Clancy said the model would be used to train agents to protect the real complex.
The request came as Mr Clancy testified before Congress on Tuesday about the agency's recent security lapses.
Construction would cost $8m (Â£5.4m) and take place at an existing training facility in the suburbs of Washington, the New York Times newspaper reported.
The Secret Service is charged with protecting the president and other high-profile figures.
It currently uses a not-to-scale model of the north grounds of the White House. That facility uses bike racks to represent fencing.
"We train on a parking lot basically," Mr Clancy told members of Congress. "We don't have the bushes, we don't have the fountains, we don't get a realistic look at the White House."
The new model would be built at a Secret Service training facility in Beltsville, Maryland, a suburb of the national capital.
The New York Times, citing anonymous officials, reported that a design for the new model had not been finalised.
Director Clancy was speaking at a hearing held by a congressional subcommittee, where he was grilled over a series of high-profile security lapses involving his agency.
During aggressive questioning, Mr Clancy revealed that several days passed before he was told about a recent incident involving two senior agents who had crashed a car at the White House after a night of drinking.
"I should have been informed," he told the hearing panel, and noted that he only learned about it from an anonymous email.
Angry that the two agents had not been dismissed, Representative David Young said: "I'm surprised that these two agents...haven't stood up and said 'I resign.' What do you do with them?"
Mr Clancy noted that the agents had been reassigned and could not be fired while an investigation is being conducted.
Six months ago, an intruder climbed over the White House fence and ran inside the facility while brandishing a knife. In April of 2012, several Secret Service agents were accused of soliciting prostitutes while on presidential trip to Colombia.
The director of the US Secret Service has asked Congress for money to build a detailed model of the White House.