In a corresponding move, Chicago recalled right-hander Matt Foster from its group at its alternate Schaumburg training facility.
Lambert, 25, made his MLB debut against the Twins on Saturday and had recorded two scoreless innings for the White Sox so far.
The White Sox selected Foster in the 20th round of the 2016 amateur draft. He had a 3.20 ERA in 43 relief appearances split between Double-A and Triple-A in 2019.