THERE IS a disturbing undertone in news accounts of the latest exchanges between the mayor of this city and the man in charge of prosecuting all local criminal cases here. It is not just that the relationship between Mayor Barry and U.S. Attorney Joseph E. diGenova is again strained and that assorted "leaks" and misinterpretations may have contributed to it. More important is the standing of the elected local government. The talk has been loose and it has reflected on Mayor Barry's reputation and on his capacity to govern. It arose from (in some cases ongoing) investigations of city hall subordinates and personal friends of the mayor. It is out of hand. For their own good as well as the city's, both men have got to cool things down.

In a front-page report yesterday, Mayo Barry directly charged Mr. diGenova with "trying so hard to discredit this administration . . . to get me into something that he never could get me into." Mr. Barry said Mr. diGenova "talks more than he produces" and voiced concern that "as long as there are suspicions . . . about massive corruption . . . it masks all the positive things we have done." He was reacting to a statement by Mr. diGenova a day earlier that the conviction of former deputy mayor Ivanhoe Donaldson had exposed "raw corruption" in the city government.

We have been critical in the past of the way information -- sometimes misinformation -- was spread about investigations involving Mr. Barry that were being conducted by Mr. diGenova's office. There is no denying the corrosive effect of rumor and gossip about charges that have not even been made. Yesterday we asked Mr. diGenova about the latest flare-up and his relationship with city hall. He flatly refused to reply to the mayor's criticisms, emphasizing repeatedly a desire "not to get engaged in a public battle. I feel very badly about it. I don't want to flame this thing up."

Mr. diGenova went on to praise Mayor Barry for announcing additional moves to monitor and investigate internal matters at city hall, terming them "excellent contributions to a battle that everybody in public office has to deal with." The prosecutor also pledged to "work with this city and with the mayor to solve any problems. It's going to be a cooperative effort, in a spirit of good faith." He insisted that he is not engaged in any campaign against the mayor or the District government, and that he was ready to be held to his word.

In the meantime, the prosecutor has more investigating to do, and that is Mr. diGenova's job. The mayor has a government to run, and that is Mr. Barry's job. Both have serious responsibilities to a public that is entitled to performance and to facts. Hot unfounded accusations about how the other is doing have no rightful place in this city's affairs.