Fontenot answered the question about possibly moving Jones broadly, but the question asked was a direct one about the team's star receiver for a decade.
"That's one of those things when you're doing things the right way in an organization, you have to listen if people call, on any player," Fontenot told Fox 5. "Especially, we are in a difficult cap situation, that's just the circumstance and it's not a surprise for us. We knew the circumstance we were in. Our administration has done an excellent job up to this point getting us in a position to be able to manage the cap, and yet we still have more work to do.
"So when teams call about any players then we have to listen and we have to weigh it and we have to determine what's best for the organization and we have to handle everything with class."
Jones is Atlanta's all-time leader in catches (848) and yards (12,896), and his 60 touchdowns rank him second behind Roddy White. Jones has played his entire career in Atlanta since being the No. 6 overall pick in the 2011 draft -- when the Falcons used five picks to trade up from No. 27 to take him.
He became an immediate difference-maker with 959 yards and eight touchdowns as a rookie. His next year, in 2012, he made the first of his seven Pro Bowls and had his first 1,000-yard season along with a career-high 10 touchdowns.
Jones, 32, led the NFL in receiving yards in 2015 (1,871) and 2018 (1,677).
"Obviously that particular player, we hold him in high regard. He's special in what he's done and what he continues to do here," Fontenot said of Jones. "But we have to consider any players if it's right for the team because we have to do what's best for this organization."
Jones' salary-cap number and cap mismanagement by Atlanta's former general manager, Thomas Dimitroff, are why the team would listen to offers for the star wideout. The cap situation forced Atlanta to restructure the contracts of multiple Falcons players this offseason, including those of quarterback Matt Ryan, linebacker Deion Jones and left tackle Jake Matthews.
The team did not restructure Jones' deal, though, with a $23.05 million cap hit and his base salary of $15.3 million fully guaranteed. If the Falcons were to trade Jones after June 1, they would be able to spread out the financial hit over two seasons instead of all in one.
The Falcons also hold the No. 4 pick in this week's NFL draft and could go a multitude of directions with the selection, from quarterback to pass-catcher to a trade down.