Morneau, who suffered his concussion while trying to break up a double play on July 7 last year, started experiencing symptoms after diving for a ground ball on Sunday and hasn't played since.
He originally had a concussion test on Monday, which showed he was mildly below baseline, and this test showed he's almost back to where he needs to be to return.
"He had improvement, but I wouldn't say it's back to normal," McWane said. "The reports that I got say that he's going in the right direction.
"The test he takes is just one tool we use to evaluate the concussion. So it's really more up to the doctor to determine when his scores are appropriate. But again, it's one tool, and we also have to go with how he feels. So once the doctors feel he's ready to return, we submit that request to the league to approve it. But we're not at that point."