One of the videos was taken on the Kilpatrick Turnpike and the other on Broadway Extension.
Both videos had been taken down from the site Wednesday night.
A link to an eBay auction of the Corvette that ended Dec. 7 was posted with the YouTube videos.
The top bid of $15,000 did not meet the reserve price.
â€œI was extremely concerned with Mr. Replogle's reckless driving and thought the YouTube posts might encourage others to do the same thing in order to garner attention,â€ patrol Chief Van Guillotte said in a news release. â€œOur roadways are absolutely no place for this type of dangerous driving behavior or stunts.â€
West said he didn't know if anyone was in the car with Replogle when he made the videos.
Replogle did not return a message seeking comment Wednesday.