A White House source said yesterday that William Bennett, director of the National Humanities Center in Research Triangle Park, N.C., is the "front-runner" for the National Endowment for the Humanities chairmanship.

"Dr. Bennett has been recommended by White House aides to President Reagan," said the source, who asked not to be named, "but I really don't know if President Reagan has signed off on it yet."

A second White House source said Bennett's nomination seemed likely.

Bennett, reached by phone last night at his home in Chapel Hill, N.C.,said he had heard nothing from the White House. "I have no idea," he said. "No one has told me. But I've been getting phone calls from reporters all day."

For several weeks now, the finalists for the NEH post appeared to be narrowed down to Bennett, 38, and Melvin E. Bradford, 47, a professor of English at the University of Dallas who has been actively campaigning for the job. Earlier this week, Bradford bluntly attacked the National Endowment for the Humanities, as run by its present chairman, Joseph Duffey, and asserted that some grants were awarded on a political basis only.

"That raised some eyebrows in the White House," said the source who confirmed that Bennett was the "front-runner," "but I don't know if it really affected his chances."

Well-placed sources on the Hill, in the administration and in the NEH have said for weeks that White House aides favored Bennett. But Bradford, an outspoken conservative Republican, marshalled strong and visible support among conservative Republicans in the Senate, which must confirm any presidential nominee for the post. Last week, 16 senators sent a letter supporting Bradford to President Reagan. Most prominent among Bradford's Senate supporters is John East (R-N.C.).

"It was very important to Sen. East to see Mr. Bradford nominated," said Thomas Farr, legislative assistant to East. Reached last night at his home, Farr said that he was not aware of the status of the White House nomination. "Sen. East has been a friend of Bradford's for many years and believes he is eminently qualified." Farr would not comment on anything that East may have done to push for his candidate. "The situation was very delicate," said Farr.

Bradford also said last night he had heard nothing. "My sources on the Hill haven't heard that the nominee will be Bennett ," he said. "I did hear a rumor, but then I heard the exact opposite rumor later."

Bennett has spent the last five years at the National Humanities Center, a nonprofit institute for advanced studies. He has been its director for the past 2 1/2 years. From 1971 to 1975 he was affiliated with Boston University, where he had been associate dean in the College of Liberal Arts, an assistant to the university's president, and had taught philosophy and law. Bennett, who is Brooklyn-born, is a graduate of Williams College and Harvard Law School and holds a doctorate in philosophy from the University of Texas.

Asked if he was a Republican, Bennett responded, "No comment. It's not relevant." However, he said he voted for Reagan. "I supported the president enthusiastically," he said.