The special counsel investigating Hunter Biden has charged a former F.B.I. informant with fabricating claims that President Biden and his son each sought $5 million bribes from a Ukrainian company — a stinging setback for Republicans who cited the allegations in their push to impeach the president.

The longtime informant, Alexander Smirnov, 43, is accused of falsely telling the F.B.I. that Hunter Biden, then a paid board member of the energy giant Burisma, demanded the money to protect the company from an investigation by the country’s prosecutor general at the time.

The explosive story, which seemed to back up unsubstantiated Republican claims of a “Biden crime family,” turned out to be a brazen lie, according to a 37-page indictment unsealed late Thursday in a California federal court, brought by the special counsel, David C. Weiss.

Mr. Smirnov’s motivation for lying, prosecutors wrote, appears to have been political. During the 2020 campaign, he sent his F.B.I. handler “a series of messages expressing bias” against Joseph R. Biden Jr., including texts, replete with typos and misspellings, boasting that he had information that would put him in jail.

Republicans pressured the F.B.I. to release internal reports after they learned of Mr. Smirnov’s claims. In May last year, Representative James R. Comer of Kentucky, the Republican chairman of the House Oversight Committee, threatened to hold the bureau’s director, Christopher A. Wray, in contempt if he did not disclose some details.

In July, after Mr. Wray complied, Senator Charles E. Grassley, Republican of Iowa released a copy of an F.B.I. record that included the false allegation without naming Mr. Smirnov, or questioning its veracity.