The promoted outfit have collected just two points from their opening seven games and lie at the foot of the Serie A standings.
A 3-1 defeat at Parma last time out was Sassuolo's fifth loss of the campaign and piled further pressure on Di Francesco.
However, general director Nereo Bonato said: "We are going forward with Di Francesco. We have to find, with him, the solutions to come out of this situation.
"We need to eliminate the mistakes that have seen us pay a heavy price so far."
Sassuolo have struggled in every department, conceding 21 goals while finding the net just five times this term.
"We have worked hard during the international break to find balance and put the players in the best condition to give their best," Di Francesco said. "We hope we can come out triumphant on Sunday."
Opponents Bologna are also feeling the heat heading into the match in Reggio Emilia.
The Rossoblu are also winless and find themselves 19th in the table - just one point better off than Sassuolo.
Their 4-1 home defeat to Hellas Verona in the last round followed a 5-0 reverse at Roma and marked their fourth loss of the campaign.
Bologna's main problem has been at the defensive end, with the Emilia-Romagna club having given away 20 goals in seven games.
Coach Stefano Pioli will be hoping to recover striker Rolando Bianchi, who has been struggling with a back injury, for Sunday's game.
Former Juventus midfielder Luca Marrone will make his Sassuolo debut in place of Francesco Magnanelli, who serves a one-match ban on Sunday.