Elliott held out for the entire preseason and is in danger of missing Dallas' Week 1 game against the New York Giants.
According to Jori Epstein of the Dallas Morning News, Jones expressed hope that there will be a resolution before Sunday's game: "It would be great—will be and would be—if before a short period of time passes, Zeke would be in that group. We'll see how today goes, we'll see how tomorrow goes, we'll see how the ensuing hours pan out."
Jones also didn't rule out the possibility of another player such as Prescott or Cooper getting a new deal before Elliott, per Epstein: "There's a chance. Because we're sure working on them. For sure."
After reportedly signing offensive tackle La'el Collins to a five-year contract extension Tuesday, Jones admitted that there is now less money to work with in terms of signing Elliott, Prescott and Cooper to new deals, according to Epstein: "This takes a nice chunk out of the pie. ... We'll see. There's no question the pie is getting smaller."
Per ESPN's Adam Schefter, the Collins deal actually created $5.7 million in salary cap space for the Cowboys thus season.
Elliott has two years remaining on his contract, while Prescott and Cooper are facing the final year of their respective deals in 2019.
While Elliott has nearly $13 million remaining on his pact over the next two years, Prescott is set to make just $2.03 million in 2019 and Cooper is scheduled to make $13.92 million this season.
Per Charles Robinson of Yahoo Sports, Elliott and the Cowboys were "progressing" Monday, but the Cowboys are struggling to find a way to make Elliott the highest-paid running back annually since they prefer not to front-load his deal.
After Dallas acquired Cooper from the Oakland Raiders last year, it went 7-2 to close out the regular season and won a playoff game, which speaks to how well the trio of Prescott, Elliott and Cooper meshed.
Overall, Cooper finished with 75 receptions for 1,005 yards and seven touchdowns last season. He was also named to the Pro Bowl for the third time in his four-year career. Meanwhile, Prescott threw for a career-high 3,885 yards to go with 22 touchdowns and eight interceptions en route to his second Pro Bowl nod.
Elliott led the NFL with 1,434 rushing yards and set a career high with 2,001 yards from scrimmage. He also set personal bests with 77 receptions for 567 yards and three touchdowns.
In the two years that have seen Elliott appear in at least 11 regular-season games, the Cowboys have won a pair of NFC East titles. When Elliott missed six games due to suspension in 2017, Dallas went 9-7 and missed the playoffs.
If Elliott does not sign in time to play Sunday, rookie fourth-round pick Tony Pollard is the top candidate to fill in, while veteran Alfred Morris will also get some run.