The 24-year-old, who is out of contract at cash-strapped Southampton, has emerged as a possible target for Neil Warnock in a bid to bolster his attacking options.
Palace finished a disappointing 15th last season, scoring a measly three goals in their last nine games.
Admitting a keen interest in the front man, Warnock claims BWP is "on the list."
"Bradley is one we have spoken about," he told the Advertiser.
"He is on our list."
TargetWarnock has also admitted an interest in signing West Ham youngster Freddie Sears.
"I've put it to Freddie that we play a lot of youngsters and give them opportunities and I feel this is the right club for him," he said.
"I'm hoping we can conclude it over the next week or two but there are three or four other clubs who have agreed financial terms who are talking to him."