Sensing an erosion of political party strength, Democratic Chairman John White and Republican Chairman Bill Brock have quietly agreed to work together to assure that the parties have a role in any further public financing of Senate and House elections, according to White.

"Bill Brock and I are looking at this," White told reporters at a breakfast meeting yesterday. But the Democratic chairman made it clear that he and his Republican counterpart have come to their joint position from vastly different perspectives.

White said he supports proposals for public financing of Senate and House elections, and wants to see the funds distributed to the candidates through the political parties.

Brock opposes proposals for public financing of Senate and House elections, White conceded -- but he added that Brock agreed that if there is to be any public financing, it should be done through the parties.

"We've talked about this informally several times," White said in an interview after the breakfast. "And we've come to an agreement . . . . I'll try to pass it and he'll try to kill it. But we've agreed that if it turns out that we are going to have it [public financing], the parties should have a role."

The agreement, White said, was that there would be "one proposal endorsed by both" party chairmen.

Proposals for the public financing of congressional elections were killed in the House Administration Committee earlier this year, and congressional sources say they do not expect any further action on the issue in this Congress.

But a committee source said that he was "intrigued by the thought that the two chairmen ahd come to any sort of agreement" and that in light of this, the committee staff might take another look at the public financing proposals.

Brock was traveling in the Midwest yesterday and could not be reached for comment on White's statement. Jack Mongoven, a spokesman for Brock, said it was his understanding that the Republican chairman still strongly opposes any extension of the present public financing of presidential elections to include congressional races.

But Brock's spokesman added that both Brock and White "have been working on a long-term study on how to revitalize the political parties, and have talked about setting up a bipartisan fund to sponsor studies on this question."

Brock's spokesman said that the Republican chairman agrees with White that "the political party structures are being eroded." The concern of both party chairmen has been that the public financing of presidential elections, by placing limits on campaign spending, have meant less emphasis on grass-roots efforts, which are the backbone of party organization strength, and more emphasis on mass communications, which do not involve party organization efforts.

Also, the vast proliferation of political action committees to channel contributions from special interests has meant a greater reliance by candidates on monied single-interest groups -- and less reliance on political parties as a source of funds and support and get-out-the-vote organization.

White and Brock have worked together on other projects recently, according to Democratic and Republican officials, although neither has been eager to publicize the fact. As recently as last Monday, according to White, he and Brock agreed on a joint position to seek to halt a proposal before Congress that would restrict the major parties' ability to take advantage of the bulk mail rate for nonprofit groups. Congress authorized the nonprofit rate for the parties last year, permitting them to mail a letter for 2.6 cents instead of 15 cents.

At yesterday's breakfast meeting, White also continued to back away from his earlier position that a challenge to President Carter by Sen. Edward M. Kennedy (D-Mass.) for the 1980 presidential nomination would be divisive and "suicidal" for Democrats. White said yesterday a Kennedy challenge need not be divisive, and that he felt it was his job to assure that there were none, even though his personal loyalty is to Carter.

"We've got two political heavy-weights, each with different political styles, who are going to go at each other toe to toe," White said. "It's going to be the greatest political fight in the history of our party."