Injury updates from White Sox GM Rick Hahn

General manager Rick Hahn had a bevy of injury-related updates about current and future White Sox to share Monday, starting with right fielder Avisail Garcia.

Hahn confirmed Garcia, who has been out since April 23 with a strained right hamstring, was starting a rehab assignment Tuesday with Class AAA Charlotte. He also said super-utility man Leury Garcia (sprained left knee) is ‘‘not too far behind’’ in terms of a rehab assignment.

Hahn also addressed some top prospects dealing with or getting over health concerns:

• Right-hander Alec Hansen, who last was seen pitching for the Sox during spring training, will make his first start of the season this weekend for Class AA Birmingham. Hansen, the Sox’ second-round pick in 2016, has been working at extended spring training after he was shut down with forearm soreness in March.

• Outfielder Micker Adolfo’s follow-up MRI exam on the sprained ulnar collateral ligament in his right elbow was encouraging enough to get him on a throwing program with Class A Winston-Salem. Adolfo has been limited to designated-hitter duty this season.

‘‘The goal is to get him back in the outfield and avoid surgery,’’ Hahn said.

• Outfielder Luis Robert, who started late at Class A Kannapolis because of a thumb injury, will move up to Winston-Salem in seven to 10 days. Robert is batting .304 (7-for-23) with two doubles, six strikeouts and one walk in seven games.

Abreu’s value

First baseman Jose Abreu entered play Monday with 25 doubles. That puts him on pace for 64, which would tie the Indians’ George Burns (1926) and the Cardinals’ Joe Medwick (1936) for second all-time.

An ongoing talking point about the Sox’ rebuild is whether Abreu, 31, will be part of it after his contract expires after next season.

‘‘If at that point — before that point — we haven’t come to terms on an extension, we certainly will have time then to revisit that,’’ Hahn said.

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Abreu is in his fifth season with the Sox. He is not only their top player, but he has had a meaningful influence on the younger players, particularly fellow Cubans Yoan Moncada and Robert.

‘‘We’ve made no secret about our affection for Jose, and it’s not just based upon the contribution between the white lines; it’s based on what he does in that clubhouse,’’ Hahn said. ‘‘I don’t think there’s a finer representative for what it means to be a White Sox or what we hope for our players to be on and off the field.’’

High on Tim

Tim Anderson’s 11 errors at shortstop — after he committed a major-league-high 28 last season — aren’t enough to deter Hahn’s faith in a player who will turn 25 on June 23.

‘‘There will be a few inconsistencies in the field from time to time or a couple of at-bats that don’t quite go the way you draw them up, but the improvements he’s shown on a nightly basis both offensively and defensively have been fantastic,’’ Hahn said. ‘‘He continues to develop along a path where you can very reasonably project him to be a shortstop on a championship club.’’