Delap was released by Stoke City at the end of the season, but after being invited to train with Burton by Brewers boss Gary Rowett, the former Republic of Ireland international has admitted that he has been impressed with the club so far.
The 37-year-old told the Burton Mail: "I've known Gary for years and he gave me a ring to come down and look at things.
"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."
Delay played for 45 minutes of Albion's friendly defeat to Ilkeston Town on Tuesday evening.