The people spoke on condition of anonymity to The Associated Press on Monday night because the agreement had not been announced.
Chicago also acquired right-handers Jeff Samardzija and Michael Ynoa from Oakland on Tuesday for right-hander Chris Bassitt, catcher Josh Phegley, first baseman Rangel Ravelo and infielder Marcus Semien. Samardzija had pitched for the crosstown Cubs before he was traded to the Athletics in July.
A right-hander who turns 30 in April, Robertson had spent his seven-year big league career with the New York Yankees. He took over from Mariano Rivera as New York's closer last season and saved 39 games in 44 chances.
New York had prepared for the possibility of Robertson's departure by agreeing last week to a $36 million, four-year contract with left-hander Andrew Miller, who joins Dellin Betances in the back end of the Yankees' bullpen. New York's offer was for far less than Chicago's, another person familiar with the talks said. That person spoke on condition of anonymity because the amount was not revealed.
Robertson's deal has an average salary of $11.5 million, falling $1 million below the average of Jonathan Papelbon's four-year deal with Philadelphia that began in 2012.
Robertson will be joining a team that finished fourth in the AL Central at 70-92, one whose saves leader was rookie Jake Petricka with 14.
The agreement with Robertson stunned at least one of Robertson's new teammates. Outfielder Adam Eaton tweeted the message "Wow," along with a photo of Sesame Street's Bert with the caption "mind blown."
Because the White Sox had among the top 10 initial picks in next year's amateur draft, they lose a second-round selection next June rather than their first-round choice, which is eighth overall. The Yankees receive an extra pick between the first and second rounds.
AP Sports Writer Howie Rumberg contributed to this report.