Chief Justice Warren E. Burger reopened his offensive against incompetent trial lawyers yesterday, only four hours after Attorney General Griffin B. Bell cast doubt on Burger's estimate that up to half of them are unqualified.

With Bell seated near him on the podium here at the mid-year meeting of the American Bar Association, Burger then departed from his prepared text to thank the sponsors of a resolution that calls on him "to publicly repudiate" the estimate or "to provide background data conclusively supporting" it.

"Whatever their intentions, they have done more for this cause than all the speeches and the lectures I and others have given for a dozen years," Burger said.

The sponsors are the Illinois State Bar Association and its president, Carole K. Bellows. The resolution is to come up today in the House of Delegates, the ABA's policymaking body.

For many years Burger has been saying that probably one-third to one-half of the lawyers who try serious cases aren't qualified to give their clients fully adequate representation. Standing "firmly" on that estimate yesterday, he said that "inadequate performance" by many trial lawyers is "one of the major reasons for congestion and delay in the courts."

To remedy the situation, Burger proposed that "special standards for admission to practice in the courts, as distinguished from practicing law generally," be required. "To treat a bare certificate of admission to practice law as a passport to try any court makes no more sense than to say that a medical school degree qualifies the holder to perform every kind of surgery," he said.

On Friday, Burger's estimate was termed "grossly disproportionate" by ABA president William B. Spann Jr. Perhaps 20 percent of trial lawyers are unqualified, he said.

Departing again yesterday from the text of his "Annual Report on the State of the Judiciary," Burger said that even an estimate of 20 percent poses "a serious problem."

Earlier, Bell was asked about the figures during and after an appearance on "Issues and Answers" (ABC, WJLA). Noting that he has no desire to get into a dispute between the Chief Justice and the ABA president, Bell told reporters: I don't think it's anything like 50 percent." As for the 20 percent estimate, he said, "I don't know where Spann" got it. In both cases, he emphasized there has been no scientific sampling that would produce a reliable estimate.

Is the issue "a tempest in a teapot?" Bell was asked. It's a tempest that "might get out of the teapot," he replied.

Burger began his campaign against incompetent trial lawyers in 1967 while a judge of the U.S. Court of Appeals for the District of Columbia. But it didn't ignite a controversy until November, when a wire service report appeared on testimony he gave earlier to a British commission.

Burger has based his estimate on conversations with other judges and lawyers. As in the past, he told the British commission that some judges put the proportion of "really qualified" trial advocates at 25 percent, while others put it at 75 percent. "Somewhere near the midway mark is probably correct," he said.

Although the Associated Press cited this material in its account, the first paragraph said that Burger had pronounced about one-half of "practicing" lawyers not adequately qualified to represent their clients. There are about 440,000 practicing lawyers, but only about 10 percent practice in court.

Burger later clarified all of this but not to the satisfaction of the Illinois bar.

In its resolution, that group charged that "the net effect" even of the new reports is that "the chief representative of the American legal system has brought the competency of the entire profession into public disrepute."

The resolution says the "available evidence suggests the opposite is true." It cites a report saying that judges of the 2nd U.S. Circuit Court of Appeals found 92.9 percent of the lawyers appearing before them to be competent in trials and appeals.

In yesterday's speech, Burger said 39 percent of the federal judges who replied to a questionnaire rated the inadequacy of lawyers appearing before them "a serious problem." The questionnaire was sent to all federal judges, of whom 81 percent responded, by a committee of the Judiciary Conference of the United States.

Burger did not say if the judges had estimated the proportion of unqualified trial lawyers.

Colorado state Supreme Court Justice Jim R. Kerrigan has predicted that the questionnaire results, when completely analyzed, will show that the federal judges rated only 6 percent of the lawyers in their court-rooms inadequate.

In making his address to the ABA, Burger departed from his eight-year policy of refusing television coverage of his remarks. He amended the ground rules this year to allow TV coverage of the first three minutes of his speech.