Jackson, who threw a no-hitter against the Tampa Bay Rays on June 25, is 6-10 with a 5.16 ERA. He has thrown seven innings or more nine times this season. Jackson made the American League All-Star team in 2009 while with the Detroit Tigers.
"I have nothing but good things to say about Edwin," Arizona manager Kirk Gibson said in New York before his team played the Mets. "We hate to lose him. At the same time, we've picked up an important part for our future and maybe it gives us some flexibility for the future to improve some things around here."
That source said the Sox are one of several teams interested in Dunn. ESPN The Magazine's Buster Olney reported earlier Friday that the Yankees and Rays also are interested.
Jackson might not be involved in any potential trade for Dunn, with the White Sox saying they might keep the pitcher for the rest of the season, according to a major league source.
And if a Dunn deal doesn't work out, the White Sox might go another direction and try to acquire another left-handed hitter, according to the same source.
Jackson, 26, would give the White Sox a veteran starter to fill out their rotation with Jake Peavy lost for the season because of a detached lat tendon.
"Edwin Jackson did a terrific job. We appreciate everything he's done for us," Diamondbacks interim general manager Jerry Dipoto said. "He threw a big no-hitter for us against Tampa Bay and was a quality individual who brought a lot to our team, but we felt like this was a great opportunity for us to build toward the future with Daniel Hudson and even further in the future with a prospect like David Holmberg."
Last week, the White Sox told Washington that they had no interest in a deal proposed by the Nationals that would have sent Dunn to Chicago for Gordon Beckham straight up, according to a major league source with knowledge of the negotiation.
Dunn, who will be a free agent at the end of the season, is batting .277 with 24 home runs and 64 RBIs, and would give the White Sox the left-handed hitting balance they have been searching for in their lineup.
Washington is looking for veteran starting pitching to go along with young phenom Stephen Strasburg.
Hudson is 1-1 with a 6.32 ERA with the White Sox after beginning the season with Triple-A Charlotte.
"They just told me that they appreciated the hard work I put in and how I handled everything that's been happening the last week and helping them to win a game," Hudson said on "The Waddle & Silvy Show" on ESPN 1000. "You can't really feel too bad.
"Obviously, you're gonna get traded for somebody and move because somebody wants you, and a new door opens up, and I'm gonna go out there and do my best."
Holmberg, 19, was 1-1 with a 4.46 ERA in eight starts with Advanced Rookie Great Falls.
The White Sox entered Friday's play 1½ games ahead of the Twins in the AL Central. Minnesota improved itself Thursday, trading for Nationals closer Matt Capps.
The White Sox recalled right-handed pitcher Lucas Harrell from Charlotte; he will make his major league debut Friday night against the Athletics.
Bruce Levine covers baseball for ESPNChicago.com. Information from The Associated Press was used in this report.