Borbon, 27, is a career .283 hitter who spent parts of four seasons with the Rangers. He was designated for assignment on April 9 and the Rangers had 10 days to try to trade him.
"It's another way of getting the organization stronger. Guy can play every outfield position," Cubs manager Dale Sveum said. "Can run, drives the ball. He's pretty much a .290 hitter with 600 at-bats in the big leagues. It's a really nice pickup. You don't have to trade or spend money. You just don't find those assets on every corner."
The move means the Cubs will play with one backup infielder right now, Cody Ransom, who they claimed off waivers on Thursday.
Borbon made the Rangers after batting .317 in 32 games in spring training. But he was the odd man out with Craig Gentry and Leonys Martin in center field. Borbon, who was selected by Texas in the supplemental first round (35th overall) of the 2007 June draft, played in just one game for the Rangers this season.
Gonzalez batted .217 in 11 games for the Cubs.