Hesitancy would be understandable. Barbosa has never once run for elected office and Brazil's treacherous political scene is hardly for beginners. But analysts say Barbosa, who rose from poverty to first black chief justice of the Supreme Court, has unique credentials.
This is a country that in just three years has seen president Dilma Rousseff impeached, her replacement and current President Michel Temer charged with corruption, and her predecessor Lula Inácio Lula da Silva jailed two weeks ago.
So in an election filled with uncertainty, the only sure thing is that many Brazilians want an outsider. And with the country's elite, including Lula, Temer and scores of lawmakers, enmeshed in the mammoth Lava Jato ("Car Wash") corruption scandal, they also want someone tough on graft.
Enter Barbosa. As a justice, his main achievement was overseeing another big anti-corruption crusade, which began in 2005, badly battering then-president Lula's Workers' Party (PT). And since retirement in 2014, he's been largely out of the public eye. In other words, he ticks both boxes.
Brazilian Socialist Party head Carlos Siqueira characterizes Barbosa as the healing figure the country needs.
"His electorate," Siqueira told AFP, "wants normality to return to Brazil."
In addition to his crime-fighting reputation, Barbosa brings an all-Brazilian story that only Lula, with his epic rise from shoeshine boy to two-term leftist president, can match.
Technically, Barbosa would not be the first black president. That honour is generally given to Nilo Procopio Pecanha, who was of mixed race and ruled from 1909-1910.
But at least in modern times, Barbosa's election would be revolutionary. Despite more than half the country identifying as non-white, there are virtually no black leaders in politics or business and racism is ever-present.
And Barbosa has already proved he can crash through those glass ceilings.mThe eldest of eight children, with a bricklayer father, his first job was as a cleaner in the labor court in the capital Brasilia.
Working gruelling hours, he studied law at the University of Brasilia, served briefly in the diplomatic service, learned several languages and then rose inexorably through the legal ranks until reaching the Supreme Court.
But if outsider status is one of Barbosa's main strengths, it also brings liabilities, including a lack of clarity about his political stance and questions over his complete inexperience in electoral politics. Barbosa is touted as the potential centrist who can conquer the space between leftist Lula and hard-right candidate Jair Bolsonaro, a former Army officer.
However, he has yet to position himself firmly on the many economic and political controversies marking that polarized landscape.
"It’s difficult to know if he'll go more to the left and try to get Lula's votes or back to the figure he was, to recall that image, where he goes more to the centre-right," said Michael Mohallem, a law professor at the Getúlio Vargas Foundation.
"He will have to decide. He can be the centre candidate, but the choice he makes might take him out of the centre."
There are also questions about whether Barbosa has the right temperament.
During his time on the court, he earned a reputation for occasional irascibility, one time upbraiding a fellow justice by warning him to remember that "when you address me, you're not talking to one of your hired thugs."
Could this former cleaner who went on to clean up the corrupt elite reach the very top? The last big name touted as the dream outsider candidate, popular TV presenter Luciano Huck, fizzled out.