Putnam, 26, is 1-0 and no earned runs allowed in four appearances with Triple-A Charlotte. He struck out 11 and walked one in six innings, holding opponents to a .182 batting average.
Signed in November, Putnam pitched in 15 major league games over parts of three seasons with Cleveland in 2011, Colorado in 2012 and the Chicago Cubs last season.
Veal was 0-0 with a 7.50 ERA in seven relief appearances with Chicago. Scott Downs becomes the inly left-hander in the White Sox bullpen.