When Senator Robert Menendez was charged last year with corruption after investigators found $486,000 in cash stashed around his house in New Jersey, he offered a simple, “old-fashioned” explanation: It had been his custom to withdraw cash from a personal savings account to keep at home, a habit he learned from his Cuban immigrant parents.

But federal prosecutors, in papers filed late Friday, presented fresh details that they suggested undercut Mr. Menendez’s claim. Some of the cash was wrapped in bands showing it had been withdrawn, at least $10,000 at a time, from a bank where Mr. Menendez and his wife “had no known depository account.” This, prosecutors said, indicated “that the money had been provided to them by another person.”

Recently, Mr. Menendez’s lawyers had asked a judge to exclude much of the cash discovered in the home as evidence when the senator’s trial in Manhattan starts next month, arguing that there was no proof the money was linked to a crime. The prosecutors’ Friday filing was in response to this request.

The issue of the cash cuts to a critical theme of the government’s case: that the senator and his wife, Nadine Menendez, had a lifestyle that was above their means and funded by bribes.

A federal indictment says that the cash, along with gold bars and other valuable items, were “fruits” of a bribery scheme. Much of the cash found in the couple’s house in Englewood Cliffs, N.J., was discovered in a bedroom closet, prosecutors said in their filing. Additional cash was found in a duffel bag in an office, in a bag on a shelf above a coat rack in the basement, in the pockets of men’s jackets hanging on the coat rack, and inside footwear under jackets. In addition, more than $70,000 was found in a safe deposit box maintained by Ms. Menendez, the government said.

Mr. Menendez, 70, and his wife are accused of accepting bribes in exchange for the senator’s willingness to use his political influence to disrupt criminal investigations in New Jersey and to help the governments of Egypt and Qatar. The Menendezes and two New Jersey businessmen charged in the scheme have all pleaded not guilty.

Mr. Menendez, the former chairman of the powerful Senate Foreign Relations Committee, and the two businessmen are expected to be tried together, starting May 6 in Federal District Court in Manhattan. The senator has refused widespread calls for his resignation and has said that he hopes to run for re-election in November as an independent if he is exonerated.

Prosecutors have said that at least 10 envelopes of cash found in searches of the senator’s house and a safe deposit box, containing more than $80,000, bore the fingerprints or DNA of one of the businessmen also charged in the case. They said gold bars had also been tied to the businessmen.

Mr. Menendez’s lawyers had no comment on Saturday. But in court papers, they have said they were trying to bar prosecutors from telling the jury only about cash that had no clear link to an alleged co-conspirator. “The government should not be permitted to try to sway the jury with dramatic presentation of valuable items bearing only a speculative connection to any charged offense,” they wrote.

The judge, Sidney H. Stein, ruled Thursday that Ms. Menendez, 57, will be tried separately, after her lawyers said she had a newly diagnosed medical condition that will require a surgical procedure and possibly a lengthy recovery. The judge set a tentative July trial date for her.

Last September, Mr. Menendez, who was born in the United States, offered an explanation for at least some of the money found in their home, tying it to his family’s roots in Cuba in the years before Fidel Castro seized control.

For 30 years, Mr. Menendez said, he had withdrawn thousands of dollars in cash from his personal savings account, keeping it “for emergencies.” He said the habit stemmed from a “history of my family facing confiscation in Cuba.”

“This may seem old-fashioned,” he added, “but these were monies drawn from my personal savings account based on the income that I have lawfully derived over those 30 years.”

The next month, in an interview with PBS, Mr. Menendez elaborated, stating that he had withdrawn $400 in cash every week for “the better part” of three decades.