Godley made his bullpen debut in Wednesday's 4-1 loss to the Pirates, allowing a run on three hits in two innings.
As for who will take Anderson's place, the D-backs are still considering their options with reliever Josh Collmenter and Minor League right-handers Aaron Blair and Jhoulys Chacin as top candidates.
If the D-backs decide to go with Blair, who was the 36th overall pick in the 2013 Draft and has had success with both Mobile and Reno this year, he would need to be added to the 40-man roster. Otherwise Blair would not need to be protected on the 40-man until after next season.
For some organizations, that would be a concern, because 40-man roster spots can be precious.
"We haven't worked here under the norm as you know, so I would say that, no, that's not a factor," D-backs general manager Dave Stewart said.
Instead, Stewart's main concern is whether Blair would be ready for the promotion, and Stewart will rely heavily on the opinions of Major League scout Mike Russell, farm director Mike Bell and Reno manager Phil Nevin.
Anderson, meanwhile, was asked to work on his fastball command. After a strong start to the season, Anderson hit a rough patch, had a brief stint on the DL and then struggled in two of three starts after being activated.
"I think he just needs a little confidence builder and go down there and work on some things in a less-pressure situation," D-backs manager Chip Hale said. "Is it going to be 10 days? I don't know. We'll see how he does down there. He still has a chance to come back and be a very effective pitcher for us."
Anderson, 27, was 5-5 with a 4.51 ERA in 21 starts this year.
Steve Gilbert is a reporter for MLB.com. Read his blog, Inside the D-backs, and follow him on Twitter @SteveGilbertMLB. This story was not subject to the approval of Major League Baseball or its clubs.