Sources tell NBCDFW.com's Newy Scruggs, no teams are expected to bite on Padilla off of the waiver wire, so he'll remain with the team.
Earlier in the week, Texas requested outright waivers on Padilla. Major league teams have until noon Friday to claim him.
If no team claims him, Padilla could be shipped to the minors, but Padilla has to agree to the assignment or he would become a free agent.
On Thursday, Major League Baseball fined Padilla for intentionally throwing at Yankees first baseman Mark Teixeira.
After the game, Teixeira had choice words about Padilla, claiming the right-hander threw at him intentionally. Padilla has a history of throwing at hitters and has hit at least 10 batters in five seasons during his career. He has already plunked seven batters this season.
This year, Padilla is 3-3 with a 5.57 ERA in nine starts.