The Falcons are aware of Jones' plans and hope he changes his mind, per sources.
Jones could be fined up to $40,000 for each day he misses. The Falcons' first practice is Friday.
The team has previously said that it will not rework his deal.
Falcons owner Arthur Blank told The Athletic that he does not plan to get involved in any negotiations over Jones' contract at this time.
"I have full faith in TD [general manager Thomas Dimitroff] and DQ [coach Dan Quinn] to handle this in the best manner that is feasible for us. We love Julio, and I have said our wishes are that he be a Falcon for life. As always, I am available to our staff as needed and at appropriate times," he said.
Jones signed a five-year, $71 million contract in 2015 that included a $12 million signing bonus and $47 million guaranteed. His base salary this season is $10.5 million.