His law firm Akin, Gump, Strauss, Hauer & Feld, grieved the loss of a "colleague, mentor and friend", who died yesterday.
"To US leaders, he was a trusted counselor and an honest broker. And to those across the aisle or the table, he was a man of integrity who always sought common ground," it added.
"His wisdom, judgement and foresight were matched only by his sense of humanity, and we are as diminished by his loss as we were enriched by his life."
A "deeply saddened" Nancy Reagan recalled that Strauss was one of the first people she and the late Republican president had met when they first arrived in Washington more than three decades ago.
"People ask me how I could be acquainted with him since he was such a staunch Democrat," Reagan recalled in a statement.
"My answer was always quick and firm -- it didn't matter to him what political party I belonged to and it certainly didn't matter to me which political convention he attended."
In 1991, president George HW Bush, also a Republican, picked the Democratic confidant to serve as US ambassador to the Soviet Union as the communist country crumbled.
Following the collapse of the USSR that year, Strauss served as US ambassador to Russia.
"Bob Strauss may have cut his teeth in the brass knuckle and highly partisan political fields of Texas politics, but he counselled several presidents of the United States of both parties -- and like the others, I valued his advice highly," said Bush, a fellow Texan.
"Bob was one of the greatest leaders the Democratic Party ever had, yet presidents of both parties relied on his advice, his instincts and his passion for public service -- not to mention his well-honed sense of humour," Obama added.