Clarke paid tribute to his predecessor after it was announced that the former Rangers, Everton and Scotland manager had died at the age of 73 following an illness.
Smith won seven of his 16 matches in charge of the national side including the famous 1-0 win over France at Hampden in 2006.
"He was a formidable manager and a formidable man," Clarke told the SFA website. "He always had a strength and presence that marked him out and no doubt contributed to the great success he had throughout his career.
"I had my first experience at the World Youth Championship when he was Andy Roxburgh's assistant in 1983 and, even back then, he was impressive in his coaching and how he carried himself.
"He was always very supportive and terrific company when our paths crossed over the years.
"His loss will be felt profoundly across Scottish football."