Lucas Giolito threw a baseball around Central Park on Monday, playing catch with pitching coach Ethan Katz. He’ll be in Queens on Tuesday night, facing the Mets’ Carlos Carrasco at Citi Field.

This will be his 161st start for the White Sox, and possibly the last. There’s wide-spread demand for starting pitching as the August 1 trade deadline approaches, and the Sox are positioned to be major supplier.

Given their 40-55 record, it would be smart for White Sox General Manager Rick Hahn to move as expeditiously as possible in trading Giolito, Lance Lynn and, if it is possible, Mike Clevinger.

Giolito and Lynn have started 35 percent of the time for the Sox over the last three season, including 38 times thus far in 2023. It’s unclear how Katz and first-year manager Pedro Grifol would fill such a gaping void, and that’s sort of the point.

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The White Sox are currently positioned to have the fifth pick in the 2024 draft. The top two picks seem to have been already determined, with the only question being whether Oakland (25-70) or surprisingly weak Kansas City (27-67) wind up with the first pick. But the Sox could wind up with the third pick if they play worse than Washington and Colorado down the stretch.

That’s an unpleasant bit of math for the fans who frequent Guaranteed Rate Field but the higher you pick in the draft, the better your chances to add an impact player. Late-season wins are empty calories for non-competitive teams.

Reports have several teams following the White Sox as they search for starting pitchers. The question is how can Hahn get the most for rental pieces.

Giolito, who is earning $10.4 million, will be a free agent after this season. He has shown that the 4.90 ERA he carried last season was an outlier, starting this season 6-5 with a 3.45 ERA in 19 starts. He’s been especially efficient over his last seven starts, holding opponents to a .182 batting average while striking out 48 in 44 innings.

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Lynn too is turning it up at the right time. He got uncharacteristically knocked around early in the season but has been missing bats of late. He has 47 strikeouts in 31 innings in his last five starts, holding opponents to a .218 batting average.

Lynn had one year left on his deal with the Texas Rangers when he was traded to White Sox before the 2021 season. Hahn extended him for two years, $38 million. That deal includes an $18 million club option for ’24, with a $1 million buyout.

Those terms could be attractive to pitching-needy teams as they give them some built-in flexibility for next season. There’s little down-side risk if Lynn does not perform well enough to justify the $18-million commitment.

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Clevinger, signed to a one-year deal with a mutual option for ’24, was an unpopular signing due to a domestic violence investigation but has pitched solidly. He’s 3-4 with a 3.88 ERA in 12 starts but is currently sidelined with biceps inflammation. The injury looked ugly when it happened on June 14 but he’s bounced back well and is currently throwing bullpens.

His position as a possible trade piece is dependent on how quickly he returns and how well he pitches. He’s earning only $8 million this year but is due $12 million in ’24 if both sides exercise his option. He is due a $4 million buyout if the White Sox or an acquiring team reject his option for next season.

Touki Toussaint, acquired on waivers from Cleveland in June, is a wild card. He’s pitched well in two starts and three extended relief outings, compiling a 3.10 ERA while striking out 20 in 20 1/3 innings and holding opponents to a .179 batting average.

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Touissant is a former first-round draft pick who failed to fulfill the high expectations placed upon him. But he’s only 27 and has three years of team control remaining. He bears watching, either as yet another White Sox starter on the trade market or, more likely, a potential long-term arm on the South Side.