The 37-year-old former Stoke man played for 45 minutes of the Brewers' pre-season defeat at Ilkeston on Tuesday night.
He will continue to train with the club this week and their location could swing it for the Derby-based defender.
"I've known Gary (Rowett) for years and he gave me a ring to come down and look at things," Delap told the Burton Mail.
"Burton would be brilliant, but it's got to be right for the club and me. I've had a few offers but none really viable thinking about my family."
Rowett added: "In an ideal world he's going to stay local. He'll have offers from one or two decent clubs but he's based in Derbyshire and at his age spending time with your kids can be as important as anything."
- Sports & Recreation