A UFC official confirmed that Evans was offered the fight in September after Quinton Jackson withdrew due to injury.
Evans has not fought since April 21 when he lost to current light heavyweight champion Jon Jones by unanimous decision.
"I will fight anybody with the proper notice," the 33-year-old Evans said on Twitter. "But after eight years with the UFC and being one of their top earners, why would I take a bad fight?
"Bad fight meaning not being in shape. Losing is hard enough but I will never serve myself! If I haven't been training I won't fight!"
Evans said on Friday that he seeks to return to action in December. He has revealed an interest in moving to middleweight, but plans to compete at light heavyweight in his next fight.
Evans is currently negotiating with UFC officials on a return date. An opponent has yet to be named.
Though his next fight will be at light heavyweight, Evans said he would like to weigh in somewhere around 201 pounds for the 205-pound bout limit.
"I'm using this fight [in December] to get my weight down and keep it under control," Evans told ESPN.com on Friday. "Afterward it might be easier for me to go down to 185."
When Evans (17-2-1) does return to action he intends to be in top physical condition. He said it's the primary reason he refuses to take any fight on short notice.
"I know a lot of you fans may not realize, but I don't walk around ready to fight. No one does," Evans said on Sunday. "In this sport it takes at least eight weeks to be ready.
"Fighting another person is much harder than most people think, and if I'm not training it takes time to get ready. There are no easy fights."
The hard-hitting Teixeira (18-2) has knocked out 11 of his opponents. But in his long-awaited UFC debut on May 26, Teixeira showed his offensive diversity by submitting Kyle Kingsbury via first-round arm triangle.