The 33-year-old Burton was 0-1 with a 4.50 ERA in four appearances this year with Class A Tampa and Triple-A Scranton/Wilkes-Barre. He was released by the New York Yankees on May 16.
Burton is 18-19 with a 3.44 ERA and 10 saves in 367 relief appearances with Cincinnati (2007-11) and Minnesota (2012-14).
Texas also said Sunday that right-hander Kyuji Fujikawa cleared unconditional release waivers and became a free agent. He was designated for assignment on May 17 and put on waivers Friday.