MINNEAPOLIS -- Strong outings by Dylan Cease and Lucas Giolito and some opportunistic offense had the White Sox in position to win two games in Minnesota against the American League Central-leading Twins.

Instead, the White Sox chance to gain some ground in the division went by the wayside as Minnesota came through late against Chicago’s bullpen again on Sunday afternoon.

The Twins scored three times in the bottom of the ninth inning to tie the game, and they won, 5-4, on Ryan Jeffers’ RBI single in the 12th to sweep the White Sox at Target Field and perhaps set up Chicago to be sellers at next week’s Trade Deadline.

“Yesterday and today, we put ourselves in position to win,” White Sox manager Pedro Grifol said. “We [could have] come in here and beat these guys two out of three. We’re leaving out of here 0-3. They all sting. When you do a good job of pitching and you do a good job of manufacturing some runs and put yourself in a position to win and then you lose it, they sting. That’s just a part of it.”

The White Sox certainly had designs on using the weekend series against the Twins to try to claw back in the race for the division. Instead, Chicago has lost 11 of 15 and fallen to 12 games behind Minnesota and fourth in the AL Central.

Giolito pitched five scoreless innings with nine strikeouts, including his 1,000th career K, but he had his pitch count rise. Kendall Graveman allowed three runs on three hits and a walk in the ninth to take his fourth blown save, while Jesse Scholtens (1-3) gave up an unearned run and two hits in the 12th.

“Want to take one here, at least,” Giolito said. “Lost the lead late. Battled in extra innings, but they came out on top. It is what it is. It sucks, though.”

With 61 games remaining in the season and the Aug. 1 Trade Deadline approaching, the White Sox have some decisions to make. One of the potential options for pitching-needy teams could be Giolito, who is in the final year of his contract and will be a free agent in the offseason.

The 6-foot-6 right-hander is in his seventh season with Chicago and has been a top-11 vote-getter in AL Cy Young Award voting three times. If teams are interested, outings like the one on Sunday would improve his stock.

Giolito bounced back from giving up a season-high eight runs over 3 2/3 innings in his last start against the Mets. He is 0-1 with a 1.00 ERA (two earned runs in 18 innings) with 23 strikeouts against the Twins this season.

Asked about the upcoming Trade Deadline, Giolito said, “I’m cool with it. Whatever happens, happens. Same thing I’ve been telling you. It’s not really on, like, the forefront of my mind when it comes to when I’m here and I’m getting ready to pitch and I go out and pitch. I’m just going to do my job just like any other year, any other day. That’s it. That’s my mentality. See what happens.”

Giolito and Cease combined to allow one run in 11 innings on Saturday and Sunday. Eloy Jiménez and Zach Remillard homered on Sunday -- with Remillard hitting his first career Major League homer -- but the White Sox couldn’t close when they had chances.

Chicago left a runner on base in six of the final seven innings, with eight total runners stranded in that stretch. The team went 2-for-13 with runners in scoring position on Sunday after going 1-for-9 in the same situation on Saturday, leaving a total of 18 men on base in the final two games of the series.

“It’s not going to cut it in a Major League game,” Grifol said. “I know hitting’s tough. We put ourselves in a good position to win this game. But just like we’ve got to win it on the pitching end, we also can’t blow it away on the hitting end. But these guys battled. It’s an unfortunate loss. This one really hurts. 3-0 lead in the ninth against these guys -- this one stings a little bit.”