Speed, 42, who began his career at Elland Road in 1988 and made nearly 250 appearances before moving on to Everton in 1996, was found dead at his home in Chester on Sunday morning.
Grayson, who signed schoolboy forms at Leeds on the same day as Speed, told the club's official website: "It is an unbelievably sad day. Speedo was a true friend and a very, very popular man.
"Our sympathies are with Louise, his children, his dad, and all his family. We signed on the same day and played in the youth and reserve teams together.
"We also did all our coaching badges together. He was a good friend and this is a very sad day.
"Gary was such a popular person and this is a hard time for everybody."