Wales football manager Speed was found dead at his Chester home on Sunday (November 27), aged 42.
Savage, who played with Speed for the national team, mourned the loss of his close friend on Twitter yesterday afternoon and on BBC Radio 5 Live's 606 show, describing him as "one of the greatest ever Welsh players".
He also revealed that he had spoken to Speed on the phone on Saturday and joked about his latest performance on Strictly.
"The world has lost a great man in Gary Speed," wrote Savage. "I'm devastated, spoke to him yesterday morning why! why, why!! I'll miss him so much x
"He came to watch Strictly 3/4 weeks ago. I high fived him in the front row he loved the show, he loved life he loved his family! Devastated.
"He was upbeat on phone yesterday we were laughing together, talking football and dancing he was a great teammate and a great friend.
"Speedo loved Strictly he was an avid fan last time I saw him. I was high fiving him from the dance floor his smile was huge!"
In an interview with the BBC, Savage added: "I spoke to him yesterday and he was in high spirits. I just can't believe it. He was my mate and he's gone.
"He's left two gorgeous kids behind. Gary was [a professional]. He was one of the fittest guys. He looked after his body. He was a great teammate, a great manager.
"He was joking yesterday, saying don't get a two off Craig [Revel Horwood]."
After co-hosting a 606 tribute special to the late football star, Savage added: "Hope I done my mate Speedo proud tonight on 606. Overcome with emotion when the anthem was being played just thinking of him! Devastated."