The American League Central-leading White Sox will begin a three-game set at the Oakland Athletics on Tuesday night, and Lambert reportedly will start the series opener.
Right-hander Matt Foster was optioned to Charlotte to make room on the roster. The reliever is 2-1 with a 6.06 ERA in 33 appearances with Chicago this year.
The 26-year-old Lambert is making his third big league start -- all coming this year. He is 0-1 with a 9.00 ERA in three appearances with the White Sox this season.