The former Peterborough United and Lincoln City boss died after returning home from seeing his side lose 1-0 to Notts County at Meadow Lane on Tuesday night.
He is believed to have suffered a brain haemorrhage.
Alexander suffered a brain aneurysm in November 2003 while in charge of Lincoln, but eventually returned to management.
He recently took time off from the game to aid his recovery and missed Macclesfield's home match with Accrington last month after suffering from severe hiccups.
The former St. Lucia international became the first black professional manager in 1993 when he originally took charge of the Imps, before later returning to Sincil Bank and leading the club to four successive League Two play-off campaigns from 2003-2006, losing twice in the final at the Millennium Stadium, Cardiff to Bournemouth and Southend.
Macclesfield chairman Mike Rance told BBC Sport: "We're all absolutely devastated.
"I spoke to Keith last night at about 11:15pm after our game at Notts County and he was in good spirits because, although we lost, we had given a team chasing promotion a good game.
"We were chatting about an FA hearing after he had a row with a referee at Bournemouth, which just shows how passionate he was about the game.
"It was perhaps fitting that the last conversation we had was about football
"He went home after the game last night and collapsed. Tragically, he never recovered.
"Keith was the perfect manager for a club like ours. He was totally pragmatic about managing in League Two and the constraints involved.
"He was a splendid man, a real gentleman and an absolute privilege to work with. He was strong, sensible, commanded respect throughout the game and was unique in many ways."
The Silkmen are scheduled to play Hereford United at Edgar Street on Saturday. It would have been Alexander's 100th in charge of the club.
However, Rance revealed that he had already contacted the Bulls' chairman Graham Turner about the news, and said that Alexander's family will decide whether the match will go ahead at the weekend.
"Graham also knew Keith very well and the decision as to whether Saturday's game goes ahead will be taken by Keith's family shortly," he said.
"I know what Keith would have wanted, but it will be his family's decision."
Macclesfield striker Ben Wright echoed the thoughts of his chairman.
"We've had a day off today and some of the lads were sat round and it just came up on the television," he told the Manchester Evening News.
"We all knew Keith hadn't been quite so well but he had looked fine in training and at the game last night. It's come as a massive shock and we can't really believe it. We're all in shock.
"I think whether we play or not on Saturday will be down to Keith's family. I spoke to the chairman briefly and he's going to talk to the players but I think what happens on Saturday will be down to Keith's family."
Hereford chairman Turner told the Bulls' official website: "We were shocked and saddend to hear of the death of Keith Alexander. Saturday's match against Macclesfield Town now becomes of small significance compared to this news.
"Our condolences go out to his family and friends and all at Macclesfield Town.
"We fully support the decision to allow Keith's family to decide if Saturday's games goes ahead."
The England team facing Egypt at Wembley on Wednesday and the England U21 side taking on Greece in a Euro 2011 qualifer at Doncaster's Keepmoat Stadium will wear black armbands in honour of Alexander.
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