Investors are betting new leadership will help restore credibility to the scandal-tainted firm and ramp up production and boost profits, traders said.
Petrobras preferred shares, the company's most-traded class jumped 15.5 percent in Sao Paulo to close at 10 reais, a five-day high. Tuesday's gain was the largest one-day jump since Jan. 15, 1999.
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Pressure has been mounting on Rousseff to clean up Petrobras, whose reputation suffered with the arrest and testimony of three former senior executives and three dozen others, including executives of major suppliers.
Police say they have uncovered a price-fixing, bribery and political kickback scheme that allegedly benefited Rousseff's ruling Workers' Party as well as others. The illegal activity, authorities allege, diverted at least $3.7 billion and perhaps more than $28 billion from Petrobras coffers.
Petrobras said last week that the corruption was one of several factors that helped wipe out a net 61.4 billion reais ($22.7 billion) from the value of its assets, such as refineries and oil platforms, but refused to take a charge against earnings.
Re-elected in October, Rousseff has stood firmly behind Foster, a long-time associate. A Petrobras employee since the 1970s, Foster moved up company ranks rapidly during Rousseff's 2003-2010 stint as chairwoman of Petrobras' board of directors.
Foster, who has publicly acknowledged that she had previously offered to resign, has said she did nothing wrong.
A separate report in Folha said Finance Minister Joaquim Levy has been sounding out potential replacements for Foster. Police plan to question a Petrobras employee on Tuesday who said she warned Foster of the corruption but was ignored, the G1 news service reported.
Rousseff's predecessor, former Brazilian President Luiz Inacio Lula da Silva, also plans to pressure Rousseff to drop Foster, according to Folha's print edition.