"We are happy to have had the chance for a short-term deal with a player like Serdar Tasci and bring him to Bayern Munich," said Bayern sports director Matthias Sammer.
A 14-time Germany international, Tasci spent the majority of his career at VfB Stuttgart, with whom he won the Bundesliga in 2007, before transferring to Spartak in 2013.
"I feel really well and look forward to this challenge," the 28-year-old said in a statement. "I hope I can help the team."
Martinez, who was not included in the squad for the 2-0 win over Hoffenheim on Sunday, is the second Bayern defender to suffer a recent injury with Jerome Boateng ruled out for up to three months with an adductor muscle injury.
Fellow central defender Medhi Benatia is still working on his comeback, making Holger Badstuber the only experienced central defender in coach Pep Guardiola's squad.
Bayern are on track for a record fourth Bundesliga title after remaining eight points clear at the top following Sunday's win. They also take on Italy's Juventus for the Champions League round of 16 later this month.
(Reporting by Karolos Grohmann, editing by Alan Baldwin)
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