LOS ANGELES -- Paul Konerko could be on a Minor League rehab assignment by later this weekend, if all goes well with his work this week.

Konerko was placed on the 15-day disabled list on June 17, with the move being made retroactive to June 15, due to a strained left oblique muscle. It was Konerko's first break due to injury in his long and storied career.

As of Tuesday, Konerko was taking dry swings. He is scheduled to hit off of a tee on Wednesday, followed by a tee and flip drills on Thursday, flips on Friday and possible batting practice on Saturday, according to the White Sox. The team entered Tuesday with a 3-3 record in Konerko's absence.

The White Sox also updated the status of veteran hurler Esteban Loaiza, who was placed on the disabled list retroactive to June 11 with a right shoulder strain. Loaiza was put on a throwing program to build up his arm strength and his velocity, which was hovering around the 80 mph mark during his short stint with the White Sox.

But Loaiza's program was changed in order to have him better geared toward relief work. Instead of having Loaiza throw three-to-five innings per appearance, he's now going to throw two innings more often. He'll have two innings and then two days off, followed by two innings and one day off.

According to reports received by manager Ozzie Guillen, the desired velocity increase from Loaiza was not present in his last appearance for Triple-A Charlotte.

"If the velocity doesn't come up, it's going to be tough for him to come up, to be honest with you," said Guillen of Loaiza, who is 0-1 with a 6.00 ERA in two games and six innings during his rehab stint. "I would like to see more velocity and see more out of him, because to be honest with you, I can't really use him.

"He needs to get better. He needs to get healthier and better. I worry about velocity and location. That's the things were looking for. It's not there yet."