Mayor Marion Barry has no plans to respond to an ultimatum by Washington Redskins owner Jack Kent Cooke that gave the District until June 30 to come up with a proposal for a new football stadium or face the prospect of losing the team to the suburbs, a spokesman for the mayor said yesterday.

"We're not issuing anything about the stadium," said spokesman John C. White. He said the mayor had not spoken with Cooke since the owner delivered the ultimatum in an interview with a local television station Wednesday.

The city is hiring consultants to explore the feasibility of building a replacement for RFK Stadium, but the study is expected to take at least six months.

Cooke said last summer that RFK, which seats 55,000, is too small and asked the city to build him a new one. He has said he wants a 75,000-seat domed arena for the Redskins, whose lease at RFK expires in 1990.

Cooke has made it abundantly clear that he does not plan on paying for a new stadium, and has held out the threat of moving to the suburbs if the city does not make a satisfactory proposal.

Wednesday, as hundreds of thousands of cheering fans packed downtown Washington to celebrate the Redskins' return from their Super Bowl victory in San Diego, Cooke told WJLA (Channel 7) that the city had until June 30 "to come up with a solid proposal."

"Failing that, it's going to be necessary for us to go to one of the surrounding counties and say, 'We've exhausted every means of keeping the stadium in D.C. {and} we are now awaiting bids from each of the counties to determine which is the best one for us to move to,' " Cooke told the television station.

Suburban officials were lukewarm to the news of Cooke's ultimatum. "We're always willing to talk with anyone, but we have no proposal to do anything," said Connie Bawcum, deputy county executive of Prince William County. "We are not pursuing the Redskins."

By and large, suburban officials have said they would not be willing to spend public money on a new stadium, as Cooke apparently expects them to do. They seem to be honoring a pledge made under the auspices of the Metropolitan Washington Council of Governments last fall to help keep the Redskins in the District.

One possible exception is Loudoun County, where a consortium of private interests has been exploring the possibility of building a stadium on its own.

The District plans to hire consultants to evaluate the possible location and design of a new stadium, as well as potential financing options. Cooke has said repeatedly that he would prefer to keep the Redskins in the city.