Washington — Republicans are trying to patch up the ''gender gap.'' At the midterm of the Reagan presidency, the GOP has seen its popularity with women sag. Mr. Reagan has won the dubious distinction of being the first chief of state ever to poll so poorly among the female population.
The most recent Gallup poll finds that only 36 percent of women, compared with 47 percent of men, approve of the President's performance. And exit polls following the last elections showed that women are turning toward the Democratic Party.
But in the last few weeks Republicans have begun moving on almost every front to improve their position with women.
* President Reagan, upbraided for naming too few women to top administration posts, made back-to-back appointments of Elizabeth Hanford Dole and former US Rep. Margaret M. Heckler (R) of Massachusetts to his Cabinet. His administration has also entered a Supreme Court lawsuit against pension plans that pay women lower rates than men.
* Sen. Bob Packwood (R) of Oregon toured the presidential campaign trails of New England with his message that the GOP must win over women and minorities. He's really not a presidential hopeful, he says, but taking that trip was the only way to get the attention of the news media.
* Republican women in the House are finding listening ears in the administration as they make a plea to spare nutrition, displaced-homemakers, and day-care programs from budget cuts. ''I think the administration is improving the course,'' says a hopeful Rep. Claudine Schneider (R) of Rhode Island, a past critic of White House policy toward women. ''This should be reflected in the budget proposals.''
The President has won widespread praise for naming Mrs. Dole to the Department of Transportation and Mrs. Heckler to head the giant Department of Health and Human Services. Both have been long established in Washington politics and have been advocates of women's rights.
The Reagan administration, with 94 women holding presidentially appointed jobs, now claims to have the best such record in history. And with three women in Cabinet-level posts, it appears to have solved the problem of having few women in visible posts.
''I'm delighted,'' says Senator Packwood, who doubts that the White House is following his advice. ''I think they're listening to Dick Wirthlin (presidential pollster) and some of the other pollsters.''
While applauding the appointments, the senator points to the administration's stand on the pension case in the Supreme Court as even more important ''for women in the marketplace,'' since it could affect the entire private insurance system. At issue is whether insurance companies and other groups can pay women lower annuities because, on average, women live longer than men.
Packwood is sponsoring legislation to make all gender-based discrimination by insurance companies illegal. The bill, soon to be reintroduced in the new Congress, is part of an economic equity package that also has the backing of other prominent Republicans, such as Sen. Robert Dole of Kansas, Elizabeth Dole's husband.
Packwood says he hopes that the White House will now adopt parts of the equity package, even if it continues to oppose the Equal Rights Amendment. ''There are still things they can do that would help mitigate the damage,'' he says.
Diana Lozano, special assistant to the President, denies that the new White House sensitivity toward women comes from poll-watching. She traces the new efforts to last August when the White House Coordinating Council on Women began meeting. Mrs. Dole, then a presidential adviser, proposed the six-person group of high-level White House staff members.
''Since that group formed, there has been a lot more momentum'' on issues concerning women, including hiring and economic problems, says Ms. Lozano. ''We hope there will be a reference in the State of the Union,'' which the President delivers Jan. 25.
Representative Schneider and fellow Republican women in the House will be looking for evidence of more sensitivity on Jan. 31 when the President's budget reaches Capitol Hill.
''These are very important steps that the President has taken,'' she says of the Cabinet appointments. ''I think that the next step needs to be to reach beyond the upper echelon to the women in the grass roots and let his budget policies better reflect a sensitivity toward women in general.''
Mary Jean Collins, vice-president of the National Organization for Women (NOW), offers no praise to the administration. ''We do see window dressing,'' she says. '' We don't see a change in the policy.''
''There is a decline in enforcement of equal-opportunity laws, a cutting of social programs of importance to women,'' says the NOW official. She points out that women have been hit by a double whammy since they are often both recipients and workers in social programs cut back by the Reagan administration. The one positive sign: ''The fact that the Reagan administration recognizes that they have a serious problem with women is good,'' she says, adding that it is further proof that women have political clout.