A proposal for a televised debate between Mayor Marion Barry and Patricia Roberts Harris won eager support from Harris yesterday, but the mayor and his backers were cool to the idea.

Officials of three private-sector employes unions that support Harris proposed a two-way debate be held a week or two before the Sept. 14 Democratic mayoral primary. They said they would pay half the $12,000 cost of purchasing one hour of prime television time for the debate if the three major public employe unions supporting Barry agreed to pay the other half.

The union's proposal would exclude from the debate D.C. City Council members John Ray and Charlene Drew Jarvis, two other Democratic mayoral candidates who so far appear to be drawing far less public support than Barry and Harris.

Harris said shortly after the plan was unveiled that she would welcome the chance to debate Barry.

"I'm prepared to discuss the real issues with the mayor, and I'm delighted that a group of concerned labor officials has made this proposal," she said. "It will undoubtedly benefit the voters of this city."

Barry said he would not participate in a two-candidate debate.

"I believe in democracy," Barry said. "The democratic process says four people are qualified to run on the ballot. I'll go anywhere, any time with all four. Why should I become undemocratic now?"

An official of the American Federation of Government Employees (AFGE), one of three major public employe unions backing Barry, rejected the proposal, saying that Barry, who leads in the polls, would have nothing to gain politically by debating Harris.

"She needs it, he doesn't," said AFGE lobbyist Bernard Demczuk. "We are not going to spend $6,000 for a debate that has been going on for free for the last three or four months."

Officials of two other major public employe unions backing Barry could not be reached for comment on the plan yesterday.

One of Barry's aides, who declined to be identified, said he considered the debate proposal a desperate attempt by Harris to turn the campaign around, but Sharon Dixon, Harris's campaign director, disputed that analysis.

"To couch it in political terms is to miss the point of what a campaign is about and what the nature of the office of mayor is all about," Dixon said. "The issue is not how best to position yourself in a political contest. The issue is to enlighten the public . . . To miss that point is to miss the point of what it is to be a mayor."

Barry has avoided one-on-one meetings with Harris, a former Carter administration official, although he has appeared on the same platform with her at scores of mayoral forums.

David L. Robinson, president of the Washington Building and Construction Trades Council, one of the proponents of the debate, said it was time for the public to see Harris and Barry debate one-on-one.

"With all due respect to the other candidates, with each month it has become more clear that this is a race between Marion Barry and Patricia Harris," he told reporters."

Robinson was joined by Ron Richardson, secretary-treasurer of Local 25 of the Hotel and Restaurant Employees Union, and Thomas R. McNutt, president of Local 400 of the United Food and Commercial Workers, in proposing the debate.

Jarvis said yesterday she would be "absolutely opposed" to a Harris-Barry debate. "It is not for the leaders of a labor organization to make a decision on who their members should listen to."

Ray joked yesterday that the debate might violate federal regulations requiring broadcasters to give candidates equal time. "I'm sure that as soon as she [Harris] finds out about it, she'll put a stop to it," he chuckled.

(An FCC official said the proposed debate would not violate fairness regulations as long as those Democratic candidates who were excluded were given the opportunity to buy an equal amount of time.)