Democratic governors, intent on using their influence to put a moderate stamp on their party's 1984 platform and campaign, are showing increasing interest in Sen. John Glenn (D-Ohio) as their standard-bearer against what they assume will be a strong reelection bid by President Reagan.

Former vice president Walter F. Mondale has more endorsements than anyone else in the six-man Democratic presidential field and is reported to be well ahead in organizational activity even in states whose governors are neutral or skeptical about his candidacy.

But interviews during the meeting of the National Governors' Association here showed a developing trend toward Glenn, particularly among southern, midwestern and western Democrats who are, in the words of Arizona Gov. Bruce Babbitt, searching "to see if a centrist candidate emerges who can be a credible alternative to Mondale."

Glenn appears to be on the verge of picking up the support of two or three important gubernatorial allies. Kansas Gov. John Carlin, picked here to become chairman of the association next summer, has invited Glenn to a major fund-raising event on Aug. 14. Carlin said today, "There's no question Glenn would be the strongest candidate in Kansas."

Virginia Gov. Charles S. Robb, chairman of the Democratic governors' caucus, though vowing to remain publicly neutral until after November's legislative elections, is described by Carlin and other colleagues as a private advocate of Glenn's cause.

Perhaps of greatest significance, Alabama Gov. George C. Wallace, a potential kingmaker in that state's key primary next March, showed a strong leaning toward Glenn in an interview.

Wallace said he has visited with Glenn more often than any of the other contenders and "he impresses me as a man who would be a strong candidate for the Democratic Party. He has the same kind of image as Eisenhower did."

Wallace also said that some of his own strong supporters, notably businessman Marvin Warner, are already working to line up support for Glenn. Despite his deafness and apparently increasing physical disability, Wallace remains a powerful political force in his home state.

Mondale can claim public support already from five governors: Rudy Perpich of his home state of Minnesota, Anthony S. Earl of Wisconsin, James Blanchard of Michigan, Harry R. Hughes of Maryland and J. Joseph Garrahy of Rhode Island. He is the early favorite in all of those states.

Each of the other contenders except Sen. Alan Cranston of California, which now has a Republican governor, can count on the formal backing of the top elected official in his home state. Glenn and Sens. Gary Hart of Colorado and Ernest F. Hollings of South Carolina and former Florida governor Reubin Askew have their governors behind them, but in several cases they still must expect a challenge on their home grounds.

The reports of Mondale's organizational aggressiveness span the country from North Carolina and Maine to Idaho and Arizona. But there was a notable hesitancy in the comments of some governors previously identified in political speculation as potential early allies for Mondale.

Massachusetts Gov. Michael S. Dukakis said he is "genuinely undecided" among Mondale, Cranston and Glenn. Connecticut Gov. William A. O'Neill said his state party is "pretty evenly split" between Mondale and Glenn. Texas Gov. Mark White said he now plans to stay neutral through that state's delegate selection contest. Arkansas Gov. Bill Clinton acknowledges inroads in Arkansas by both Glenn and Cranston.

With 34 Democrats now in governorships and party rules giving them more sway on their delegations than at any time in the past three elections, they are clearly conscious of the clout they can wield.

At their closed breakfast meeting today Democratic National Committee Chairman Charles T. Mannatt said he wanted them not only to head their delegations to the San Francisco convention but to run the party registration drives and the presidential campaign in their states.

Mannatt promised them two representatives on each of the major convention committees, including the platform-drafting unit, and a slot for a "major policy speech" to the convention. Support is developing among the governors for New York Gov. Mario M. Cuomo as the convention keynoter.