SÃO PAULO—Marcelo Odebrecht, the jailed former head of Brazilian construction giant Odebrecht SA, on Thursday agreed to sign a plea-bargain agreement in connection with Brazil’s largest corruption probe ever, according to a person close to the negotiations.

The move could roil the nation’s political class yet again. The testimony of the former industrialist, which is part of the deal, has the potential to implicate numerous politicians who allegedly took kickbacks from contractors as part of a years-long graft ring centered...