That all changed abruptly.
The big tight end visited the Oakland Raiders and many expected him to sign there. But then came word that he had been diagnosed with a herniated disc and Gresham underwent surgery in mid-March.
There still was a market for the 6-foot-5, 260-pound player, but at a greatly reduced value.
But Gresham wound up with a one-year deal with the Arizona Cardinals. Details weren't revealed, but it's safe to say it's far less than he would have received had the injury not been detected.
''I'm not going to lie to you, it was pretty tough,'' Gresham said. ''I had a lot going on this offseason, and at the end of the day I have peace with playing football. I came here to play football. All the other stuff, it was tough, but it's all behind me now and I'm looking to the future.''
He said Arizona's need for a tight end wasn't the reason he came to the Cardinals.
Apparently quarterback Carson Palmer's persistent urging was. The two played together in 2010, Gresham's rookie year and Palmer's last season with the Bengals.
''He did a good job recruiting me,'' Gresham said. ''He stayed on my phone a lot, just giving me reminders and he sold it well.''
Palmer downplayed his influence on Gresham's signing with Arizona, but praised the tight end's abilities.
''He's a mismatch against linebackers,'' Palmer said. ''He can be a mismatch depending on the safety. He's really, really good in the run game and can pass block against defensive ends, which is rare for tight ends. He's just a big, physical presence on the field.''
In his one season with Palmer, Gresham caught 52 passes for 471 yards and four touchdowns. A first-round draft pick (21st overall) out of Oklahoma, Gresham had 280 catches for 2,722 yards and 24 touchdowns in his five seasons with Cincinnati. Last season, he caught 62 for 460 yards and five scores.
Gresham once had a reputation for dropping passes, but his statistics aren't particularly bad in that category. His most were seven drops in 2012. He dropped three in 2013 and two last season. Fumbles have been something of a problem. He has three in each of the past two seasons.
But he's a welcome presence for a team of young tight ends, and one of them, Troy Niklas, is out with a hamstring injury.
Gresham said he's ''just trying to get in there and fit in.''
As for when he will be able to play, Gresham says only ''soon.''
Coach Bruce Arians at least wants to have Gresham removed from the physical unable to perform list so he can participate in walk-throughs.
''We'll hopefully take him off the list he's on so he can do the walk-throughs and start getting the reps and gradually work him into football shape, because it's taken him awhile to get in shape,'' Arians said before Sunday's practice.
''It's not different than when we got John Abraham (just before camp began in 2013). John was not in the best football shape. We'll take our time with him, but we need to get him out there and start getting reps mentally, which he can do in our walk-through.''
Palmer said it will take some time for Gresham, even when he's healthy enough to practice.
''This is a whole new world for him. He's got a lot of catching up to do,'' Palmer said. ''He's got really kind of a crash course in this offense. He's got his work cut out for him.''
Notes: Three players left Saturday's first workout with injuries. Rookie outside linebacker Shaq Riddick injured a hamstring and was to undergo an MRI. Inside linebacker Sean Weatherspoon aggravated a hamstring injury and took Sunday's practice off. Arians said he didn't think it was serious, calling it a ''wily old veteran move'' in order for Weatherspoon to be fresh when the team practices for the first time in pads on Monday. ... Outside linebacker Darryl Sharpton had a ''very minor'' hip flexor, Arians said.
AP NFL websites: http://www.pro32.ap.org and http://www.twitter.com/AP-NFL