The Democratic Party's bid to regain control of the Senate in 1986 and win the White House in 1988 focused yesterday on the search for a new national party chairman, with indications that Democratic governors intend to have a strong voice in the selection.

Charles T. Manatt said he took the job as chairman of the Democratic National Committee four years ago with the presumption that he would serve one term. But he said he would make a decision by the end of the month after conferring with the state chairmen and congressional leaders.

Govs. Bruce E. Babbitt of Arizona and Charles S. Robb of Virginia began a search for a new national chairman before last Tuesday's elections, a spokesman for Babbitt said. Tops on their list is outgoing Gov. Scott Matheson of Utah, a former chairman of the Democratic Governors' Association, and former U.S. representatives Bob Kreuger of Texas and John Cavanaugh of Nebraska. All three are reported to be seriously considering.

Babbitt and Robb also approached Gov. James B. Hunt Jr. of North Carolina, who lost an expensive and hard-fought race Tuesday to Sen. Jesse Helms (R-N.C.), and retiring Sen. Paul E. Tsongas (D-Mass.) but the spokesman for Babbitt said both had taken themselves out of consideration.

Another possibility for the post is Rep. Tony Coelho (D-Calif.), chairman of the Democratic Congressional Campaign Committee, who has been praised because the Democrats were able to limit the Republicans' gains in the House Tuesday, which could frustrate their hope of reconstituting the working ideological majority they enjoyed in the House in 1981 and 1982.

Party rules specify that the job be full-time, and there is speculation that the Democrats might create a general party chairman post similar to the one Sen. Paul Laxalt (R-Nev.) holds in the Republican Party.

Other possibilities for national chairman are Paul Kirk, a former aide to Sen. Edward M. Kennedy (D-Mass.); Robert Keefe, a party consultant who is former political director of the Democratic National Committee and ran Sen. John Glenn's presidential campaign; former California Democratic chairman Nancy Pelosi; Texas Democratic chairman Bob Slagle; Duane B. Garrett, a major fund-raiser and national campaign co-chairman for Walter F. Mondale, and Sharon Pratt Dixon, a former member of the DNC executive committee from the District of Columbia.

The first round in the fight will take place Nov. 16-17, when the state chairmen hold their winter meeting in the Virgin Islands. The DNC, which will elect the new chairman, holds its regular winter meeting at the end of January.

Because of Mondale's landslide defeat in the presidential election and the Democratic Party's strength at the state and local levels -- Democrats hold 34 of the 50 governorships and, according to Manatt, 60 percent of the nation's elective offices -- there is strong sentiment for getting the governors more heavily involved in the national party.

Selection of a western governor such as Matheson also would dramatize the desire of some leaders to change the perception that the party is a Washington-based apparatus dominated by organized labor, congressional leaders and New Deal liberals.

"Many of the governors want a national chairman who would be an effective spokesman for the party in the near future," said Jim West, a spokesman for Babbitt. "Elected officials such as Matheson, Cavanaugh and Kreuger have been through the wars. They've had to stand up to their constituents and to the press."

AFL-CIO officials have complained that there was no visible Democratic Party presence in the field during the campaign and that their financial contributions to the state parties were wasted.

The labor federation wants a "good administrator and a good spokesman," one official said. John Perkins, head of the AFL-CIO's political arm, said it would "hang loose and see who the players are" before backing a candidate.

AFL-CIO President Lane Kirkland intends the organization to make a preprimary endorsement of a presidential candidate in 1988 as it did for 1984, according to spokesman Murray Seeger. Perkins said 62.7 percent of labor-endorsed congressional and gubernatorial candidates won on Tuesday, compared with 59.5 percent in 1980.

Manatt and his Republican counterpart, Frank J. Fahrenkopf Jr., chairman of the Republican National Committee, each held news conferences yesterday to contend that Tuesday's elections gave his party the base to dominate for the rest of the century.

Fahrenkopf said the Republicans registered 4.1 million new voters in 1984 and that an ABC News exit poll indicated that 5 million more voters identified themselves as Republicans than in 1980.

ABC exit polls showed that 32 percent of the voters identified themselves as Republican compared with 28 percent in 1980. Democrats were down slightly, from 40 percent in 1980 to 38 percent, while independents were about the same, 28 percent in 1984, 27 percent in 1980.

"I'm tremendously excited about 1984," Fahrenkopf said. "Young voters supported President Reagan and registered Republican by 2 to 1."

He said that the Republican Party was building its county organizations with the aim of controlling a majority of the state legislatures by 1991, when they will redraw U.S. House districts after the 1990 census. The Republicans won control of both houses of the Connecticut state legislature and control of one house of the state legislature in five other states on Tuesday.

"We plan to get Hispanics and blacks increasingly involved in the local party structure," Fahrenkopf said. "We think Jesse Jackson's candidacy was a very good thing in getting blacks excited and involved in the process but this election showed them the danger of aligning themselves with one party and shutting the other one out."

Manatt responded that the young voters' support of Reagan was not a permanent factor. "It's like 1956 when we were all concerned with job and economic security and we had a lovable grandfather in the White House," he said, in reference to President Dwight D. Eisenhower. "It's anything but permanent.