Angelo Wright, Jackson's agent, told The Associated Press on Monday that "We think it will be tomorrow."
Wright said he is confident Jackson will continue playing, no matter the result of the appeal with NFL officials.
"I'm not worried about it either way," Wright said, adding he plans to "file something" if the appeal is denied.
Jackson, Minnesota Vikings DTs Kevin Williams and Pat Williams, New Orleans Saints running back Deuce McAllister and defensive linemen Will Smith and Charles Grant and Houston Texans long snapper Bryan Pittman reportedly tested positive for the diuretic Bumetanide. The drug helps rid the body of excess water and salt but is considered a masking agent for steroids by the NFL.
All of those players appealed. Pat Williams is also represented by Wright.
The players claim the banned substance was not listed as an ingredient in StarCaps, an over-the-counter weight-loss pill.
Jackson filed suit against StarCaps in Alameda County Superior Court in California last month, seeking restitution for any lost salary and damages for "false advertising and unfair business practices."
Falcons coach Mike Smith declined comment Monday when asked if he is prepared to be without Jackson for the team's final four regular-season games.
"We can't really talk about that," Smith said.
New Orleans offensive guard Jamar Nesbit already served a four-game suspension, but he sued to recoup $235,294 in lost salary from the drug company.
The makers of StarCaps have suspended shipments of the drug and said the company was "taking all necessary steps to ensure that our customers receive product that is safe and effective."