President Reagan made it official yesterday at a White House luncheon that Frank Hodsoll, a deputy to White House chief of staff James Baker, would be the president's nominee for chairman of the National Endowment for the Arts.

However, Reagan did not announce any nominee for chairman of the National Endowment for the Humanities, as originally planned. "He hasn't made his final decision," Hodsoll said later.

The process of nominating an NEH chairman has been marked by politicking on the part of candidates, special interest groups and senators, although sources say the White House should be ready to make an announcement within several days. Sources also say the choices have been narrowed to two candidates, one of whom is Professor Melvin E. Bradford of the University of Dallas. Bradford, considered very conservative, has strong support in the Senate, which must confirm the nomination. However, he faces strong opposition from both liberal and conservative quarters on and off the Hill.

Reagan also received the final report of the Presidential Task Force on the Arts and the Humanities, presented to him by one of the task force co-chairs, Hanna Gray, president of the University of Chicago. In thanking her, Reagan said, "And I will read it."

Reagan, who proposed 50 percent cuts in the budgets of both endowments, told the luncheon guests that "American support for arts and humanities has come mostly from the private sector." The endowments, he added, "have catalyzed additional private support." He noted that "our goal is to strengthen that public and private partnership . . . Our cultural institutions are an essential natural resource. They must be kept strong."

The president's remarks were greeted with applause, but also doubts. "He has the right spirit," said author Barbara Tuchman. But she added, "I think they're naive in feeling they can turn it responsibility for funding for the arts all over to the private sector."

The luncheon group included two past NEA chairs (Nancy Hanks and Kennedy Center head Roger Stevens), the present NEA and NEH chairs (Livingston Biddle and Joseph Duffey, respectively), task force members, the directors of the large museums in town, and congressional arts supporters such as Rep. Sidney Yates (D-Ill.) and Sen. Claiborne Pell (D-R.I.).

Missing from the group were the two other task force co-chairs: Daniel Terra, ambassador-at-large for cultural affairs, absent for personal reasons, and actor Charlton Heston, who is in Vancouver making a movie. Hanna Gray read a statement from Heston, which said in part: "I've been preaching the convictions of artists. Now I'm trying to practice them."