Giggs and Speed became good friends during their long club careers and international appearances as Wales team-mates.
"I am totally devastated," Giggs said. "Gary Speed was one of the nicest men in football and someone I am honoured to call a team-mate and friend.
"Words cannot begin to describe how sad I feel at hearing this awful news. It goes without saying my thoughts are with his family at this tremendously sad time."
Former Reds captain Gary Neville also paid tribute to the former Leeds United, Everton, Newcastle United, Bolton Wanderers and Sheffield United midfielder.
"It’s absolutely devastating," Neville said on Sky Sports. "Our careers crossed paths often and I played against him many times. He was a fantastic professional, a winner with a great reputation.
"We sometimes think of football being important, but it’s not."