McDaniels worked for the Patriots from 2001-08 and returned to New England in 2011 after stints in Denver and St. Louis.
Ben Volin, the Patriots writer for the Boston Globe, said an interview would have to be completed before the end of Sunday's NFL wild-card game action. He said it is unclear if the Patriots have granted permission for McDaniels to interview with other teams.
Cleveland’s front office fired Browns coach Rob Chudzinski Sunday night just hours after a loss in Pittsburgh ended his first season with a seven-game losing streak and a 4-12 record.
McDaniels is from Ohio, but he rejected similar interest from the Browns this time last year.
NFL.com's Albert Breer was skeptical that McDaniels would leave, saying on Twitter: "A big reason McDaniels went back to Patriots was to create stability for his family. He has 4 young kids, is paid well, likes New England."
Tom Brady told WEEI on Monday he doesn't want McDaniels to leave.
"I hope he doesn't go anywhere," Brady said. "I'd love for him to coach me the rest of my life."