White House press secretary James S. Brady is making what his pleased doctors call "extraordinary progress" toward an "essentially normal" life, despite the bullet that irretrievably smashed more than a fifth of his brain.

"We are very optimistic," his George Washington University Hospital surgeon, Dr. Arthur Kobrine, said yesterday, in an almost entirely rosy assessment of Brady's condition three months after he was shot March 30 by the gunman who wounded President Reagan.

"We plan if all goes well" to let Brady go "on an outing," a few hours' auto ride within the next few days, Kobrine said, then start "sending him home for some weekends" sometime this month. The aim is to release Brady from the hospital by the end of the summer, the surgeon reported.

Brady, whose job is being held open by the White House, remains a severely wounded man "still in the relatively early post-injury phase" of neurological recovery, a kind of recovery that typically takes a full year, Kobrine emphasized in an interview.

"We're still not out of the woods," he added. The brain tissue contaminated by bullet fragments could still harbor silent forcuses of infection that could break out anytime. "Once we make it six or eight months down the pike, I'll feel a lot easier," he said.

A neurosurgeon accustomed to seeing brain-injured patients make slow, bit-by-bit recoveries, Kobrine described the improvements he has been seeing in Brady as "dramatic."

His physical condition, his ability to move and walk, remains more seriously affected than his mental condition, Kobrine said. Mentally, "I think he gets more and more like the old Jim Brady every day."

Brady's left arm and hand and his left ankle and toes remain affected -- his main injury was in the right side of the brain, which controls the left side of the body. He cannot use his left hand but he can move his left leg forward and back. He has one-hour sessions of physical therapy twice daily, walking and practicing standing and balancing between parallel bars.

"He still can't walk without the support of the bars," Kobrine said. "But I have little doubt now that ultimately he will be walking with a cane, maybe with a little brace on for a while. He will certainly walk enough to function normally, probably with a little limp."

Mentally, Kobrine said, "His intellect, his intelligence and memory and wit and humor and personality are pretty darned close to normal, and he continues improving. I'd say he's normal the overwhelming portion of the time. Sometimes he gets pretty tired and doesn't do quite as well, but that's to be expected at this point."

As to facial movement, "it's essentially normal," and "his speech is fine, absolutely normal." Normally loquacious, Brady until recently was abnormally brief, the surgeon said, "but that's changing every day, and conversationally he's improved dramatically in the last three weeks. Before, he answered appropriately but didn't initiate much. Now he will sit and kid around for quite a while."

After his injury Brady suffered a series of complications: air reaching his brain, abnormal air passages leaking brain fluid, the danger of a blood clot in his leg reaching some vital area. These seem to be gone.

The "limiting factor" in his going home to stay, said his doctor, is now simply his inability to move freely enough from bed to wheelchair to car to travel back and forth to the hospital for what will in any event be months of daily physical therapy.

"As soon as we see that he can do this," said Kobrine, "he should be able to go home."