While Twins players wait to see if Chief Baseball Officer Derek Falvey and General Manager Thad Levine add to the roster before the July 31 non-waiver trade deadline, they also know there’s another side to consider.

That someone’s job could be on the line.

“That’s out of your hands,” reliever Ryan Pressly said. “You can’t control that. They have to make this team better. They are going to do whatever they can to do that.”

When the Twins closed in on a deal Thursday for Atlanta lefthander Jaime Garcia — a deal that the Braves pulled out of — that put lefthander Adalberto Mejia and righthander Kyle Gibson on the hot seat. One of them would likely be out of the rotation to make room for him.

A deal could still happen as the sides remain in contact. Minor league prospect Nick Burdi was set to head to the Braves before they backed out because of medical concerns, but the Twins could replace Burdi with another prospect. Other teams are now talking to Atlanta about Garcia.

“I know that each time I go out there I have to give the team a chance to win,” said Gibson, who went a season-high 7 ⅓ innings on Saturday. “If they do bring in a starting pitcher and I’m the guy who is out, they will give me a chance in the pen or whatever they need to do.”

The Twins also are expected to look for bullpen help, which could affect Pressly, who is 1-2 with a 6.82 ERA in 31 games.

Gibson and Pressly talked about their situations during batting practice last week. They concluded that the best way to approach the deadline is to force the Twins to trust them.

“We can be a couple of guys that, if we can get back to throwing the way we normally can throw, we can maybe eliminate the search,” Gibson said.

Breslow let go

The Twins designated lefthander Craig Breslow for assignment. They have 10 days to trade him, release him or send him to the minors.

Breslow expressed an interest in hooking on with another major league team. Twins manager Paul Molitor, who praised Breslow for his influence on the younger pitchers, said the club will try to trade him. Breslow, 1-1 with a 5.23 ERA, gave up an RBI single and walked in a run on Sunday.

“I feel like I’ve got some outs left in me,” Breslow said.

Righthander Alan Busenitz, with a 1.78 ERA at Class AAA Rochester, is Breslow’s likely replacement. Busenitz posted a 2.08 ERA over six outings with the Twins in June before being sent down.

Hughes has tests

Righthander Phil Hughes last week had an MRI of his neck and shoulder area as doctors try to determine why he’s still showing symptoms of thoracic outlet syndrome.

Hughes is on the 60-day disabled list, meaning his season is likely over. On Tuesday, he will spend most of the day at the Mayo Clinic in Rochester, undergoing various tests.

Hughes, 4-3 with a 5.87 ERA, made nine starts this season before landing on the DL with right arm soreness. He came off the DL and joined the bullpen. But the symptoms continued.

Etc.

The Twins traded minor league righthander Nick Tepesch to Toronto for cash considerations. Tepesch made one start for the Twins, failing to get out of the second inning against Boston on May 6.