Hendriks is scheduled to throw live batting practice to teammates Friday and an assessment will be made from there on when he would pitch in a major league game.
Hendriks, 34, was diagnosed on Jan. 9 and completed his final round of chemotherapy April 5. He began a rehab assignment on May 5 at Triple-A Charlotte, where he posted a 10.80 ERA in six appearances.
"When he throws [Friday], we'll wait and see the outcome, how he feels, what the numbers were [and] evaluate it really well," White Sox manager Pedro Grifol said. "He's close, so now we've got to really do a good job of just evaluating him, how close he really is to being Liam Hendriks, not how close he is just to pitch here in the big leagues."
Grifol already said that he does not expect Hendriks to appear in a game before the end of the weekend.
Hendriks recorded 75 saves over his first two seasons with the White Sox, making the All-Star team both years. His 38 saves in 2021 led the American League and he followed up with 37 last season.
Hendriks, a native of Australia, also was an All-Star in 2019 when he was a member of the Oakland Athletics.
In 471 career games (44 starts) for the Minnesota Twins (2011 to 2013), Kansas City Royals (2014), Toronto Blue Jays (2014-2015), Athletics (2016 to 2020) and White Sox, Hendriks has a career 3.81 ERA with a 31-34 record and 115 saves. He has 724 strikeouts against 158 walks across 645 innings.