Ross, who started and homered in the Braves' 6-3 loss to the Cardinals in the National League Wild Card game Friday, is expected to return.
"Rossy has been a big part of this team for the last three or four years since he's been here," general manager Frank Wren said. "He's been a real valuable guy in a backup role. We hope to have him back."
Should Moylan return, he would be the third-longest tenured Brave. He has been in Atlanta for seven years, trailing only catcher Brian McCann and right-hander Tim Hudson.
After spending this season working his way back to the Major Leagues following shoulder surgery last September, Moylan said he is looking forward to going into the offseason healthy and is hopeful he will be able to return to the Braves.
"I really love the guys here," Moylan said. "I love everything about this team. I'd love the chance to come back here."
For some of the pending free agents, their value goes beyond how they perform on the field, extending to their presence in the clubhouse.
First baseman Freddie Freeman said that with a good, young core of hitters that includes himself, right fielder Jason Heyward and shortstop Andrelton Simmons, having veteran players helps in the clubhouse.
"We've got a lot of the guys coming back. A lot of big clubhouse chemistry guys are up for contract," Freeman said. "Hopefully they bring them back, because we would miss them."
Not long after telling reporters how much he hoped to see Hinske and others return next year, Hinske approached Freeman with a bat to sign. As Freeman wrote a lengthy message to his friend in silver marker on the black barrel, Hinske explained how much he enjoyed his three years in Atlanta.
"They're all my friends," Hinske said. "I was a big part of this clubhouse for three years. Sad to be going, but that's part of the game and business. Just look forward to the next chapter. Wherever I'll be next year or whatever."