Jones said a heavily-scrutinized Hardy, who has had a growing list of transgressions since joining the team in March amid the controversy of a domestic violence incident with a former girlfriend, must meet certain expectations before anything further is discussed.
Hardy's latest incident was being late and missing all of the morning meetings last Thursday, three days before the 10-6 loss to the Tampa Bay Buccaneers.
The missed meetings had no impact on Hardy's playing time against the Buccaneers. Hardy started at right defensive end Sunday. He played 60 of 71 snaps, making one tackle and recording no sacks.
Still, Hardy's list of distractions seems to be growing.
Hardy's late arrival Thursday came a day after he briefly changed his Twitter profile claiming his innocence in a domestic violence incident in May 2014. He was initially found guilty by a North Carolina judge, but then asked for a jury trial. Charges were eventually dropped because his ex-girlfriend refused to cooperate with the district attorney.
But even as Jones seemed to look at Hardy in a harsher light, he also continued to dilute issues involving Hardy, such as those involving tardiness and distractions during practice last week.
"But this is pro football, and the way if people's phone goes off in the meeting, if they're late for a meeting, if they don't come to a meeting, there is a fine list and we've had players fined, numerous players fined, 24-25 years out there," Jones said.
"That has been going on. So, I don't want to single him out because I couldn't tell you if last week we had four or five other people late that might get fines. Sometimes those fines, it'll be addressed, but those fines won't follow for a couple of weeks."
Hardy was suspended for the first four games of the season because of the incident for violating the NFL's personal conduct policy.
Head coach Jason Garrett acknowledged last week he talked to Hardy about his Twitter bio. The Cowboys have had to address at least four off-field incidents with Hardy since they signed him last spring.
The Cowboys remain in support of Hardy being on the team.
But a commitment for the future is not as resolute as it was just a month ago when Jones and vice president Stephen Jones said signing Hardy to a contract extension was a real possibility.
When asked about a contract extension for Hardy on Tuesday, Jones spoke hypothetically.
"Greg knows what's expected and nobody is more aware of the scrutiny and nobody is more aware, now ever more so, of what we expect of him," Jones said. "The good news is we go forward. We see if we get what's expected of him, which I fully anticipate you would get. But we see how and what's expected of him and we go from there.
"We don't, and you know this, just the nature of this one, we don't as a practice, discuss detail of extending or not extending. I'll give an indication, but that's about it. So we'll leave it for that. I have no reason to think that Greg won't do what's expected of him throughout the year."