"LeBron makes things happen pretty quickly, so we're going to have to get to it," Haden said Friday on NFL Network's "NFL Total Access."
"I say it's going to be a tight fight. I know me and [Johnny] Manziel will be pushing to make that happen before. That's why LeBron is coming here, to bring a championship to the city, and I'm trying to do the same thing."
The city of Cleveland hasn't been home to a major professional sports championship since the Browns won the championship in 1964. With the return of James to the Cavaliers and the draft-day addition of Manziel to the Browns, though, buzz is bigger in "Believeland" than ever.
"It's a great day in Cleveland," Haden said. "It's been almost overwhelming. I was actually having my football camp when people that were at my camp -- peoples' parents -- were like, 'Oh my goodness, LeBron's here. LeBron's here. LeBron's coming back home.' I'm super excited. I've been a really, really big LeBron supporter."
Haden joked during the show that he'd recruit James, a former high school receiver at St. Vincent-St. Mary in Akron, Ohio, to be a two-sport star.
"If he wants to come [play for the Browns], I'd welcome him with open arms," Haden said. "He might be the only person I could give 23."