The White House yesterday gave Vice President Bush a mild boost in his campaign for the Republican presidential nomination while proclaiming that President Reagan remains "strictly neutral" in the race.

Late in the day the White House press office released a photo showing Reagan and Bush together at their weekly luncheon meeting in the Oval Office. Officials said they took the action not as a favor to Bush but because of press requests for it.

But a White House source, who spoke on condition of anonymity, said the photo was released because "the vice president's people" wanted to balance a Reagan letter sent to rival Sen. Robert J. Dole (R-Kan.), praising him for his leadership role in helping push the administration's contra aid package through the Senate. The Senate vote was largely symbolic because the package already had been rejected by the House.

Dole has quoted from Reagan's letter to him in an effort to demonstrate that he is valued by the president, who remains popular in New Hampshire. Dole has also released a packet of similar letters from Reagan dating to 1981.

"What you have is two pro forma events, and both candidates are trying to make something of them," said a White House official. "Reagan sent letters thanking every congressman who voted for contra aid. They are a routine formality."

The official said the Reagan-Bush photo, which is not usually released, also was routine. He said it was issued to make sure that "Bush obtained an equal advantage from formalities."

Bush, who has repeatedly invoked Reagan's name in his New Hampshire campaigning, flew here for the weekly luncheon. Afterward, when reporters shouted a question about whether he would win next Tuesday's primary, Bush gave a thumbs-up sign. Asked if Dole would win, the vice president responded with a thumbs-down signal.

White House spokesman Marlin Fitzwater declined comment on the luncheon discussions and reiterated that "we're strictly neutral" in the Republican presidential race.

But some intimates say the president privately favors his loyal vice president even though Reagan has no intention of becoming involved in the primary. Since Dole won the Iowa caucuses Monday, the president has been kept away from reporters as part of what some officials acknowledged was a deliberate effort to prevent him from expressing an opinion on the results.

Officials said the current isolation of Reagan resulted from efforts by White House chief of staff Howard H. Baker Jr. to keep the White House from becoming entangled in the Republican race. Baker, a former Senate colleague of Dole, is said to believe that White House neutrality is vital because Dole's help will be needed on a number of key issues in Congress.

"But we may be overdoing it when we don't allow a question asked at a photo opportunity with the Boy Scouts," acknowledged one official, referring to a Tuesday picture-taking session that was closed to the press.