"Padilla had a personal matter he had to take care of today," Washington said. "He left this morning. We'll see the next few days how serious it is."
Washington will now push Padilla's start back to Tuesday and move right-hander Doug Mathis up a day to start Texas' series opener with Cleveland on Monday.
If Padilla is gone longer than expected he can be placed on the bereavement list, allowing the club to recall left-hander A.J. Murray from Triple-A Oklahoma to start in Padilla's place on Tuesday. Padilla would be able to remain on the bereavement list for three to seven days.
"We don't want to go there unless we have to," Washington said.
In 12 starts this season, Padilla leads Texas in wins with a 7-2 record and has the lowest ERA among Rangers starting pitchers at 3.67. In 73 2/3 innings this season, he's allowed 79 hits and 28 walks, while striking out 54.
Mathis has started two games this season for Texas and is 0-1 in those starts, having allowed seven earned runs in just 8 1/3 innings on 15 hits and seven walks.
Murray started in his only appearance for the Rangers this season, lasting 5 1/3 innings at Oakland on May 3. He earned a victory in that start, recording two strikeouts and holding the Athletics to three runs, two earned, on six hits and two walks. In nine starts for Oklahoma, Murray is 2-2 with a 4.00 ERA and 36 strikeouts in 45 innings.
Shawn Shroyer is an associate reporter for MLB.com. This story was not subject to the approval of Major League Baseball or its clubs.