Pete's representatives have not denied the singer is planning to run the grueling race on April 13, saying: "There is some truth to this - you never know what could happen."
The 28-year-old star, who is battling heroin drug addiction, has also been seen meditating and doing press-ups in a bid to clean up his act.
The source added: "He thinks that running is a great way to get back in shape - especially after piling on the pounds in rehab.
"He knows that people will be surprised by his efforts to be healthier but he really wants to turn over a new leaf."