The League of Women Voters will give independent presidential candidate John B. Anderson until Sept. 10 to achieve a 15 percent rating in the national polls and thus qualify for this year's first league-sponsored debate among presidential candidates, Anderson said yesterday.

This is later than previously had been assumed, and was taken by the Anderson camp as good news, particularly on a day when new national polls released by the Gallup and Harris organizations showed his popular support shrinking.

However, there were continuing signs that Anderson still cannot find a prominent Democrat to run on his ticket for vice president, and he has poromised to name one by Sept 1. If he fails to do so, it appears likely that his candidacy will fizzle before the League of Women Voters' deadline.

Anderson's efforts to produce a running mate of stature apparently have run up against the political ambitions of the people he has approached. Gov. Hugh L. Carey of New York -- Anderson's first choice, according to some sources -- has decided to stick with the Democratic ticket. Mayor Kevin White of Boston has said he wants to stay in the Democratic Party.

Anderson discussed the vice presidential choice with two Democratic members of the House yesterday during a visit to Washington whose full purpose was not explained.

The candidate cut short a day of campaigning in New England to fly here for a two-hour visit. He saw Reps. Les Aspin (D-Wis.) and Shirley Chisholm (D-N.Y.), and discussed a lawsuit with his lawyer that his campaign hopes will improve its chances of obtaining federal funding.

Afterward, both Aspin and Chisholm declined to endorse Anderson or say what their intentions were for the fall campaign.

Whether Anderson also had contact with some potential running mate while he was here could not be determined.

Aspin said he urged Anderson to pick former Wisconsin governor Patrick Lucey as his running mate. Lucey worked for Massachusetts Sen. Edward M. Kennedy's presidential campaign, and resigned as a delegate to the Democratic National Convention after President Carter was assured renomination.

Reached by telephone yesterday in Madison, Lucey said, "I wouldn't give a categorical no, certainly," if Anderson did ask him to run for vice president. "I guess I'd want to talk to him about it," Lucey said.

He confirmed that he had declined an offer from Anderson last week to become cochairman of his campaign with Mary Crisp, the former Republican Party vice chairman. Lucey said he has not endorsed Aderson for president, but did not rule that out.

Whether Lucey would bring any political advantage to the Anderson ticket is problematical. He is not nationally prominent, and is not popular today among leading Democrats in his own state. He resigned as governor to become Carter's ambassador to Mexico, then quit that post to help in the Kennedy campaign.

The League of Women Voters' willingness to wait until Sept. 10 to decide whether Anderson is a serious presidential candidate deserving of a place in the debates was conveyed in a letter that Anderson released yesterday. The first debate is set for Sept. 18.

David Garth, the New York-based political consultant who has played a key role in the Anderson campaign, predicted last night that Anderson would bounce back in the polls once the glow of last week's Democratic convention and "$20 million worth of free television time for Jimmy Carter" has worn off.

Noting that GOP presidential nominee Ronald Reagan had plummeted in the national polls in the last couple of weeks, Garth predicted that Carter "will do exactly the same kind of dive" soon, to Anderson's ultimate benefit. Garth also predicted that Anderson would come up with a "strong" running mate.

In three polls taken after the Democratic convention, Anderson scored 13, 14 and 17 percentage points -- all lower than before the convention.

Before flying here yesterday, Anderson spoke to the American Legion convention in Boston, criticizing Reagan's call for U.S. military superiority, and warning against the "dangerous hysteria" of a new arms race.

"We must question Mr. Reagan's claim that building more weapons will give either side nuclear superiority," he said.

Such an advantage, he add, "is simply not attainable, because each side will match the other missile for missle, warhead and launcher for launcher."

Anderson got a friendly reception. He told the convention that Reagan "would take us in the wrong direction, but President Carter has taken us in circles . . . . Weak and vacillating leadership has confused our allies and emboldened our adversaries."

Anderson criticized a recent presidential directive that put increased emphasis on targeting nuclear missiles on military targets in the Soviet Union instead of population and industrial centers.

He said the directive came "at the worst possible time, in the heat of an election campaign, when it is certain to be perceived as politically motivated."