They face 90 days to six months in jail and/or a $1,000 fine when they are sentenced Sept. 18, Forbes said.
As part of the plea deal, prosecutors will recommend that both men receive credit for time served.
Jones is making strides in his recovery, and entering treatment is helping to end his "revolving door" of drug use and jail time, Forbes said.
Jones was also arrested for driving under the influence in 2008 and possession of heroin in 2011.
"He's doing better than he has in a long time," Forbes said. "You can look in his eyes and tell that. It's a result of the judge letting him do this treatment instead of jail."
Mayor Jones said in a statement after his son's arrest that incarceration would save his life.
"If in jail or prison, I know that Zac has a better chance of living than on the outside," Jones said. "This is because Zac is a hopeless drug addict who has broken the heart and the will of everyone and anyone who has tried to help him."
Reach Travis Crum at travis.c...@wvgazette.com or 304-348-5163.