Last week, Browns vice president of football operations Sashi Brown said the team won't make a "panic" decision on Manziel's future.
A source told ESPN's Dan Graziano that new coach Hue Jackson made it clear when he interviewed that he wanted the Browns to move on from Manziel.
But Haslam said he believes the relationship with Manziel is fixable.
"Oh, yeah, I don't think there's any question about that," Haslam said Thursday night at the Greater Cleveland Sports Awards. "We talked to Johnny before he left. I know a big deal was made that Hue hasn't called Johnny yet. Well, we've got 53 players on the active and 10 more and there were a couple of other real prominent players that he just talked to in the last day or two, so I'm sure he'll get around to talking to him.
"I think we made clear, I don't have any different feelings than we had on Sunday, Jan. 3. I think Johnny made progress on the field last year. I think that's undeniable. We have a certain expectation for our players and that includes Johnny and he's got to live up to those expectations."
Jackson told Cleveland.com this week that he has reached out to a number of players, but that Manziel wasn't "in the first batch.''
The Browns selected Manziel with the No. 22 pick in the 2014 draft. Cleveland has the No. 2 overall pick in the upcoming NFL Draft.
"Look, one of the reasons the Browns are in the condition they are is they've not drafted well," Haslam said. "Our predecessors didn't draft well, and we didn't draft well. Any time a No. 1 draft pick or any high draft pick is not successful and is not a big contributor, that's disappointing to the organization."
Meanwhile, legendary Browns running back Jim Brown said he hopes Manziel stays with the team.
"I know a lot of kids that if you don't know how to deal with them you lose them," Brown said at the Greater Cleveland Sports Awards gathering. "They don't come ready-made.
"But on the other hand, the argument is that he's a man. He should know how to conduct himself. He has an an opportunity. We've got all of that. But I like him and I hope that they are able to relate, and I think if they're able to relate that he will respect the coach. I don't want to see him traded or anything like that. I just think that's never the answer to anything."
Manziel reportedly was in Las Vegas on the Saturday night before the Browns' final regular-season game despite being in the NFL's concussion protocol that sidelined him for the game against the Pittsburgh Steelers. He missed his mandatory concussion treatment on gameday at the team's facility and has been fined.
Manziel was videotaped drinking, including on Christmas Eve, during the season. He spent 73 days in inpatient addiction-rehab center in the offseason.
Manziel, 23, finished his second season with 1,500 passing yards, seven touchdowns and five interceptions in nine appearances (six starts) and a passer rating of 79.4. He rushed for 230 yards.