The deal is worth up to $4.2 million, prorated for days spent in the big leagues, a source told ESPN's Jeff Passan.
Cueto, 36, started 21 games for the San Francisco Giants last season and went 7-7 with a 4.08 ERA in what was a resurgent campaign for the two-time All-Star.
"He's a pitcher as opposed to a thrower. It's an exciting move," White Sox manager Tony La Russa said Tuesday. "We're looking forward to it. Anxious to get him here and then Charlotte (Triple-A). It's a very positive move for us."
The righty fills an immediate need for the White Sox, who recently lost starter Lance Lynn for more than a month due to a knee injury.
Cueto holds a career ERA of 3.45 across 14 seasons in the majors.