Alexander, who is 45 tomorrow, underwent tests before an operation to remove the fluid was undertaken. More tests are expected today to assess the situation.
A hospital spokesperson said: "Keith is in a stable condition at the Royal Hallamshire Hospital and has undergone a minor procedure today on the ward. This has been to drain some fluid and help relieve pressure from around the brain.
"He is to undergo more tests tomorrow and we'll know more after those tests when a decision will be made on any further surgery and treatment. He is expected to be in hospital for at least a week.
"Keith's family have asked us to thank all fans, friends and supporters from across the country for the best wishes they have received."
Alexander become a cult hero with the Town fans during his two year spell in the late 80's/early 90's. One of Alan Buckley's non-league signings, Alexander quit his job as a plumber to play professional football with Town following a £11,500 move from Barnet.
His giant frame brought a more physical side to Buckley's style of play but Alexander was no slouch on the ball. A wholehearted player who will no doubt fight this problem with the same passion he did those Fourth Division defenders.
Get well soon, Keith!