ARLINGTON, Texas — Lucas Giolito’s transformation from perhaps the worst starter in baseball to a likely All-Star is one of the best stories in the game in 2019.

Are 14 starts enough to get the White Sox thinking about locking him up with a long-term contract?

Giolito said the Sox haven’t broached the subject.

“No, that hasn’t been talked about at all,” he told the Sun-Times on Sunday. “I know that’s kind of like the new thing, organizations coming to young players. But I haven’t thought about it myself, and I don’t know where the White Sox are at, either. Just keep playing, play well, and all that stuff takes care of itself.”

Giolito, 24, who will make his first Fenway Park start Monday, is giving the Sox some bang for the buck, earning $573,000, a shade above the major-league minimum. He won’t be eligible for arbitration until 2021 and free agency till 2024, so it might behoove the Sox to get more certainty that Giolito can sustain what he’s doing.

He looks like he has figured something out, though, and the Sox set a tone by signing pitchers Chris Sale and Jose Quintana to long-term deals before they hit free agency. They also signed Tim Anderson to a similar deal before his arbitration years and Eloy Jimenez to a six-year, $43 million deal with two additional club option years this spring before he played his first game.

Giolito, who received a $2.925 million signing bonus from the Nationals when they drafted him 16th overall in 2012, is rolling with the flow. He knows it’s early in the game.

“If you do your job and play with confidence, you don’t have to worry about money because good players get paid,” he said. “When you’re talking large dollar amounts, there is always a time and place to discuss that and listen and figure that out. Probably not during the year.”

The Sox’ time is coming, Giolito said, and he wants his place to be with them.

“I love the organization,” he said. “Obviously with the opportunities they gave me from the get-go, sticking with me after last year, I have a lot of loyalty to the White Sox, and I’m excited with what we’re doing now and what we’ll be doing in the future. We’re just scratching the surface, and more guys are on the way. I’m having the time of my life and want to see this team go.”

This and that

Dylan Covey (shoulder inflammation) threw 30-35 pitches in a side session.

‘‘A rehab assignment is possible,” manager Rick Renteria said. ‘‘We’ll see how he feels, and we’ll make a determination when we get to Boston.’’

Add Kelvin Herrera to the Aaron Bummer (1.88 ERA, .146 opponents’ average) fan club: “Next year, he will be an All-Star. I guarantee it,” Herrera said.

to the (1.88 ERA, .146 opponents’ average) fan club: “Next year, he will be an All-Star. I guarantee it,” Herrera said. Anderson turned 26 Sunday.

James McCann (two singles, walk) has reached base in 11 consecutive games.

(two singles, walk) has reached base in 11 consecutive games. Leury Garcia extended his hitting streak to eight games.

extended his hitting streak to eight games. Carson Fulmer pitched a perfect inning with two strikeouts in his first appearance since his call-up Wednesday.

On Deck: Sox at Red Sox

Monday: Lucas Giolito (10-2, 2.74 ERA) vs. Eduardo Rodriguez (8-4, 4.71), 6:10 p.m., Ch. 9, 720-AM.

Tuesday: TBA vs. David Price (4-2, 3.39), 6:10 p.m., NBCSCH, 720-AM.

Wednesday: Reynaldo Lopez (4-7, 6.23) vs. Chris Sale (3-7, 3.59), 12:05 p.m., NBCSCH, 720-AM.