The Serie A new boys made a poor start to life in the top flight, failing to win any of their first seven league fixtures, but a run of just one defeat in their following seven games lifted them up the table.
However, since drawing 2-2 at Cagliari on December 1 Sassuolo have lost six league matches and won only once, to raise questions about Di Francesco's future.
Sunday's 3-1 defeat at fellow strugglers Livorno leaves Sassuolo languishing in the relegation zone and Bonato revealed the club will consider if Di Francesco is the right man to keep them in Serie A.
Bonato told Sky Sport Italia: "Considering the difficult moment the team is in, the club is taking a moment to reflect on the situation and consider how to go forward.
"We will confront ourselves with the technical staff and then make our decisions on how best to proceed. I think we will take the night to examine the situation and make our decisions in a calm manner."
Bonato did give Di Francesco some hope, however, dismissing suggestions that the coach would definitely pay the price for their poor run of forn.
"I wouldn't say that," he added. "Obviously it is a difficult situation, as we have lost five of the last six games, so we must see whether we have what it takes to turn this around."
