Falcons general manager Thomas Dimitroff confirmed that the two teams have asked for permission to interview Mularkey.
"The Denver Broncos and Cleveland Browns have requested permission to speak with offensive coordinator Mike Mularkey for their vacant head-coaching positions. The Falcons have granted permission to Mike to interview for the jobs later this week," he said.
The Broncos announced via their official Twitter account Tuesday that Mularkey would be interviewed Friday in Atlanta. The Falcons are off this week after earning a first-round bye in the playoffs.
Mularkey, 49, has head-coaching experience. He spent two seasons in Buffalo, going 14-18 in 2004-05 before resigning in 2006. This season, his offense, led by Falcons quarterback Matt Ryan, is ranked 15th in passing and 16th overall.
John Elway will be introduced as the Broncos' new chief football executive this week, and he'll interview Denver interim coach Eric Studesville for the full-time job next week.
Studesville was promoted from running backs coach when Josh McDaniels was fired a month ago and went 1-3 as interim coach. Broncos chief operating officer Joe Ellis praised his performance under trying circumstances, saying Studesville "lived up to our expectations. He did a fine job."
Studesville was moving out of his head coaching office Tuesday -- but only because Elway is moving in Wednesday. In an interview with The Associated Press in that office, he said he'll return to his old, smaller digs down the hall at the team's Dove Valley headquarters "until we figure out how everything plays out."
Although he had a four-week audition for the full-time gig, Studesville isn't exactly an open book.
"The thing I have to do going into the interview is demonstrate to them what my plan is going forward. I was thrust into this and it was more reactionary, just trying to get done what we needed to get done to prepare our team for football games. But this plan is more what I see going forward," Studesville said.
Mularkey joins two other candidates being considered for the Browns' vacancy.
Shurmur, 45, just finished his second season running the Rams' offense, a West Coast-style system favored by Browns president Mike Holmgren that sometimes became too conservative for St. Louis fans. The Rams gained just 184 yards on Sunday night, when they were beaten by Seattle for the NFC West title.
Shurmur was instrumental in developing rookie quarterback Sam Bradford, and the Browns want their next coach to do the same with their rookie, Colt McCoy, who went 2-6 as a starter after being thrust into the job due to injuries. Holmgren said McCoy's continued growth will be a priority when he selects his next coach.
"It's important," he said. "If Colt is the guy and you see the potential there, who the head coach is, who the offensive coordinator is, who his position coach is they're important for any team. But if, all of a sudden, you have the quarterback you think can be the quarterback, a young man who can be the quarterback for the next 10-15 years, hopefully it becomes even more important. Absolutely that is going to be one of the considerations in the search."
Shurmur's late uncle, Fritz, worked as a defensive coordinator under Holmgren. Also, Shurmur and Holmgren share the same agent, Bob LaMonte.
Shurmur is also very familiar with Browns general manager Tom Heckert. They worked together in Philadelphia, where Shurmur was quarterbacks coach for seven seasons.
Information from The Associated Press and ESPN.com's Pat Yasinskas was used in this report.