According to sources, Costa's injury will not require surgery but he could miss two to four weeks, which would put his availability for the Sept. 11 season opener in New York against the Jets in jeopardy.
Costa did not practice Thursday after suffering the injury in Wednesday's practice.
Before getting hurt, Costa continued to work with the first team the past two weeks -- even with the return of Pro Bowler Andre Gurode, who had knee surgery in June. Coach Jason Garrett said Thursday discussions regarding who would start at center had not taken place yet, but he added Costa could be in the mix.
It is possible Costa or Gurode could move to left guard and the other would remain at center. However, the Cowboys also like how rookie seventh-round pick Bill Nagy, who will make his second start of the preseason against the Minnesota Vikings on Saturday, has performed the past two weeks at left guard.
Veteran guard Montrae Holland is expected to be back to practice next week after missing time with a back injury. He has stepped up his conditioning and rehab work the past few days, but he would have little time to get ready for the preseason finale on Sept. 1 at Miami.
Todd Archer covers the Cowboys for ESPNDallas.com.