The National Education Association, one of the nation's most politically potent unions, has set Oct. 3 as the date for deciding which Democratic presidential candidate it will support.

To court that support, the candidates must run a gantlet of issues that includes videotaped "auditions," interviews with NEA officials that the union could use as campaign material.

However, NEA leaders are hard put to convince many that they are not committed already to former vice president Walter F. Mondale, with whom NEA has close, longstanding ties.

The October deadline puts NEA, which has endorsed candidates in the past two presidential elections, about two months ahead of the AFL-CIO, which hopes to endorse a candidate for the first time and has scheduled it for December. AFL-CIO officials acknowledge, however, that they are considering moving their decision up to early October. NEA is not a member of the AFL-CIO.

"Unfortunately, the presidential derby is under way already," said Ken Melley, NEA's political director, referring to assorted Democratic presidential candidate "cattle shows" such as the recent California Democratic Party convention and one in Massachusetts set for April.

The primaries and caucuses that begin early next year are not popularity contests, however, they are organizing work, Melley continued.

"It is the best organized candidate who is really going to steal the bacon in those early weeks," he said. That is where the shoe leather of a union like NEA comes in handy.

With 1.7 million members, which makes it the largest teachers' union in the country, NEA boasts one of the best grass-roots political organizations in the country, with hundreds or thousands of members in every congressional district. They tend to be articulate and prominent in their communities.

An October decision would give the NEA the three or four months required to gear up machinery to help its candidate or candidates in the early primaries and caucuses, which are crowded into a smaller time span than before, Melley said. Some state or local organizations may back different candidates than the national association.

An early decision date is seen as a boost to Mondale, the Democratic front-runner. The sooner the endorsement is made, the less chance other candidates will have to head him off.

The candidates who court NEA support will be required to fill out questionnaires on their stands on key issues, prepare a videotaped "show-and-tell" and submit to a videotaped followup question-and-answer session with NEA's president, NEA officials said.

NEA leaders are dedicated to helping defeat President Reagan, whose administration they have denounced as a threat to education. But, if the president decides not to run, NEA may support a Republican candidate as well as a Democrat through the primary season, Melley said.

"Howard Baker, for one, has been a stalwart for us," he said of the Senate majority leader who is considering a presidential run.

NEA membership is 40 percent Democratic, 30 percent Republican and 30 percent independent, Melley said, "and we have to show that the bipartisan option is available."

In 1976, for the first time, NEA endorsed a presidential candidate--Jimmy Carter--and sent a large number of delegates to the Democratic National Convention. In 1980, it was the largest bloc represented at the convention, with 10 percent of delegates and alternates. About one in every seven Carter delegates was an NEA member.

Still, NEA, like the rest of organized labor, has problems delivering members' votes. It endorsed Carter early in his 1980 reelection campaign, but its poll after the election disclosed that 41 percent of its members voted for Ronald Reagan.

Although NEA officials emphasize their desire that the candidate selection process be open and democratic, they are passing the chance to let the 9,000-member representative assembly decide which candidate to support when it meets in July at the annual summer meeting.

"That is too early," Melley said.

Instead, NEA's political action committee (NEA-PAC) will screen the candidates and make a recommendation. Then, when it meets Oct. 1, the board of directors will vote on an endorsement, which requires 58 percent approval.