Upon being contacted by university officials, Carroll turned them down, according to the report.
A report surfaced last weekend that former NFL coach and current NFL Network analyst Steve Mariucci was interested in the Razorbacks' job, though Mariucci said he was surprised by the news.
Mariucci apparently has Carroll's support.
"I still love the Arkansas fight song... But Mooch is the man... Go Razorbacks!" Carroll tweeted Wednesday.
Carroll has guided the Seahawks to back-to-back 7-9 records, including an NFC West title in 2010 and subsequent playoff victory. He has a proven track record of success in the college ranks, having amassed a 97-19 record in nine seasons at USC and winning two national championships.
Arkansas fired Petrino last week after it learned he lied about a motorcycle crash involving his mistress, who was an employee of the football program.