The vote was delayed on Monday after a councilman objected to how quickly the city was trying to push through the deal for the cages.
City Manager Chris Brady said the city considered the cages part of ongoing discussions with the Cubs to modernize spring training facilities.
The Cubs, meanwhile, are considering potential new homes on the Gila River Indian Community or near Naples, Fla.
If the team stays in Mesa, it wants more modern facilities than Hohokam Park or Fitch Park, where Mesa plans to install the semi-enclosed batting cages.
The Associated Press contributed to this report