An administration handyman who signed on as Nancy Reagan's chief of staff 24 days ago, announced yesterday he is resigning to join the consulting firm of another former Reagan aide, Michael K. Deaver.

Lee L. Verstandig, 48, who left his post as undersecretary of Housing and Urban Development in mid-January, said he told Mrs. Reagan yesterday that he wanted to go back into "the private sector." White House watchers believe Deaver was instrumental in getting him the job with Mrs. Reagan. Verstandig said Deaver offered him a position with his business Thursday night.

"It's a wonderful opportunity, and Mrs. Reagan was pleased that I'm going to work with Mike. I assured her I'd be of help to her whenever I'm needed," Verstandig said last night.

A White House announcement of Verstandig's departure said, "Mrs. Reagan is grateful and most appreciative of the service Lee has given not only to her but also to the president."

Word that Verstandig was quitting surprised some officials on the president's West Wing side of the White House. When he took over the job, it was expected Verstandig would be valuable in improving Mrs. Reagan's image, especially in expanding the substantive aspects of her campaign against drug abuse.

The first lady hired him last November to replace chief of staff James S. Rosebush who suddenly announced that he was resigning. Rosebush had been a deputy assistant to the president, but Verstandig came aboard as assistant to the president. He said at the time "maybe that adds some stature in terms of relations with the West Wing."

Mrs. Reagan's East Wing staff has never graduated beyond a stepchild relationship in the White House staff family.

Verstandig is no stranger to the White House. He was working there last year as assistant to the president for governmental affairs when Donald Regan arrived from his job as treasury secretary to become the president's new chief of staff. Regan brought his own people with him and Verstandig's job was targeted for someone else.

Verstandig was less than speedy in vacating his post, reportedly much to the chagrin of Regan. Eventually Verstandig was sent to HUD as undersecretary.

Verstandig's return to the White House last month surprised many on the president's staff. However, Verstandig said yesterday that he and Regan "never had any disagreements."

The choice of Verstandig as Mrs. Reagan's chief of staff was in line with his apparent role as sometime administration trouble-shooter. President Reagan sent him to the Environmental Protection Agency in 1983 when the director, Anne Burford, first got into hot water.

"The president asked me to go over and help Anne Gorsuch [Burford] at EPA after she resigned," Verstandig said, "and they made me acting administrator. So I was the poor person who had to reorganize that place, fire some people and help get Bill Ruckelshaus confirmed [as Burford's successor]."

In 1981 Verstandig went to work for then-secretary of transportation Drew Lewis, mapping his legislative strategy.

Yesterday, Verstandig said the Deaver firm does "strategic planning and a lot of my experience for the last 10 years has been related to that. After 10 years in government, I decided that 1986 was when I wanted to go back into the private sector."

Verstandig says he told Mrs. Reagan that last fall.

Calling his job with her "the shortest assignment I've had," he said he will stay on as chief of staff to the first lady until a successor is found, and he expects that will be within 30 days.