Shanahan, who was hired this offseason as the Browns' offensive coordinator, said in a recent interview with USA Today Sports that Manziel's skill set is "very similar" to Griffin's.
"Johnny and Robert are very similar," Shanahan, who spent the last four years as the Washington Redskins' offensive coordinator, told USA Today Sports. "They're both talented guys who can make plays with their legs.
"But there comes a time in the NFL when they're going to keep you in the pocket, and you're going to have to have that ability to make the throws with your arm and mind. And Johnny, just like Robert, has that arm talent and intelligence to do that."
Griffin thrived during his 2012 rookie season under Shanahan, passing for 3,200 yards and rushing for 815 while leading the Redskins to the NFC East title. The 2011 Heisman Trophy winner struggled last season while recovering from major knee surgery.
According to Shanahan, Manziel, who won the Heisman in 2012, has the skills to be as successful in the NFL as Griffin.
"You have to have some special magic, swagger to you that you can get the job done," Shanahan said. "And Johnny demonstrated that as good as anyone who's ever played college football. And we think he'll be able to carry it over.
"He's got the arm talent, and he ran around, made a ton of plays that won him the Heisman. You hope he can do that in the NFL. But you also know he won't be able to do it as much. ... Yet he still has to be him and make those plays."
Manziel, who was selected with the 22nd overall pick by the Browns in this month's draft, is expected to challenge Brian Hoyer for the starting job in Cleveland.
"You put a scheme in that everybody runs," Shanahan said. "But there's things that Johnny does better than Hoyer, and things that Hoyer does better than Johnny."
Browns head coach Mike Pettine has acknowledged that Manziel struggled earlier this month at the team's rookie minicamp. But Shanahan thinks Manziel has a legitimate chance to open the season as Cleveland's starter.
"Perception becomes everyone's reality, but Johnny has been here working his tail off day and night, doing everything we ask," Shanahan said. "He's really attacked it every day, done a good job even as each week gets harder as we throw a lot more at him.
"I've been very impressed. ... If he keeps working like he has been, Johnny should be feeling comfortable by the time we get to training camp."
The Associated Press contributed to this report.