Former vice-presidential candidate Geraldine A. Ferraro said today that she would not run for a U.S. Senate seat here because of "delays" in a Justice Department probe of her finances.

Meanwhile, after hinting for five months that he would challenge incumbent Arlen Specter (R-Pa.) in a Senate primary next year, Pennsylvania Gov. Richard L. Thornburgh (R) is expected to announce Thursday that he will not run.

Ferraro's announcement ended months of speculation about whether she would seek the Democratic nomination to challenge incumbent Republican Sen. Alfonse M. D'Amato, who has raised $6 million and is considered the front-runner.

Ferraro, whom polls showed to be D'Amato's strongest potential opponent, appeared tense and sad at a Queens news conference, reading a brief statement and halting questions after 10 minutes. The 16-month Justice probe is reportedly focusing on financing of her 1978 congressional campaign and failure to list some financial transactions on disclosure statements.

"The House Ethics Committee and the Federal Election Commission both looked into these matters in 1984 and found that I have never violated any public trust," Ferraro said. "From the outset, I have cooperated with the Justice Department and provided everything they've asked for. I've tried to get this thing moving every way I know how but to no avail."

She added, "I have absolutely no doubt that I will be vindicated."

Brooklyn District Attorney Elizabeth Holtzman, once considered a likely candidate, bowed out of the race recently. Other possible contenders are Mark Green, a writer and former employe of Ralph Nader who ran unsuccessfully against Rep. Bill Green (R-N.Y.) in 1980; Arthur Leavitt, president of the American Stock Exchange, and John Dyson, former head of the state power authority.

Democratic Party officials declined to comment on Ferraro's apparent suggestion that "delays" in the investigation are politically motivated. Had she run, Ferraro would have been "within striking distance" of D'Amato, said David E. Johnson, executive director of the Democratic Senatorial Campaign Committee.

A New York Daily News poll last month showed D'Amato leading Ferraro, 52 percent to 36 percent. But a Ferraro strategist said today that her advisers felt strongly that D'Amato would be vulnerable if her legal problems were behind her. "The issue is going to be Al D'Amato and his record," Ferraro said. "The problem was, if this investigation were still pending, the issue would be Geraldine Ferraro."

She said her decision was not influenced by Gov. Mario M. Cuomo's statements this week that he probably would not raise money or campaign for her because he would be busy with his own reelection. She said she was "taken aback" by the statements and that Cuomo called to tell her that he had been quoted out of context.

The Justice Department would not comment on the probe, but sources there said it is not unusual for cases involving prominent figures to take a long time. They said the case is being handled by career attorneys and would not go to a grand jury until a decision is made by senior department officials, including political appointees.

Thornburgh, who is entering the final year of his second term and cannot succeed himself, is to issue a news release Thursday and will not be available to answer questions, a spokesman said. Most observers have concluded that he will not run.

Thornburgh commissioned a poll that showed him 14 points ahead of Specter in a 1986 primary and winning handily against any Democrat. But Specter had put out the word that, even though Thornburgh might beat him, the primary battle would be so divisive that Thornburgh would be damaged goods.

Thornburgh, sources said, was swayed by personal considerations. "I think he was convinced he could win, but I'm not sure he's ever wanted to be a senator," said a source close to him.