Hunt received a call from the Colts before he set foot on the field to work out for the Saints and was offered more money, a source told ESPN's Dianna Russini.
The Colts have an immediate need at running back as star Jonathan Taylor and the team are in a standoff over a contract extension. Taylor left training camp to rehab an ankle injury off-site, a source told ESPN's Stephen Holder on Tuesday.
Hunt and free agent linebacker Anthony Barr met with the Saints Tuesday.
The Saints are in the market for running back depth as starter Alvin Kamara is set to serve a three-game suspension for violating the NFL's personal conduct policy. The Saints have Jamaal Williams and rookie Kendre Miller on the roster but recently placed reserve running back Eno Benjamin on injured reserve with a ruptured Achilles.
The Saints' linebacker depth has become thin this week as well, with starter Demario Davis currently sidelined with a calf injury. The Saints lost reserve linebacker Andrew Dowell for the season after he tore his ACL earlier in camp.
Saints coach Dennis Allen said he does not expect Davis' injury to be long term.
"We'll monitor him and see how it's going. I'm not going to get into any timelines, but, again, it's not going to be anything that's really significant," Allen said Sunday.
Barr, 31, spent the 2022 season with the Dallas Cowboys after eight seasons with the Minnesota Vikings. A 2014 first-round pick by the Vikings, he started 14 games last season with 58 combined tackles and a sack. He has made four Pro Bowls, most recently during the 2018 season.
Hunt, 28, was with the Cleveland Browns for the past four seasons and played in 17 games with no starts last season as a backup to Nick Chubb. He carried the ball 123 times for 468 yards and three touchdowns. He also had 35 receptions for 210 yards and a touchdown.
Hunt was a 2017 third-round pick by the Kansas City Chiefs and spent the first two seasons of his career there. He won Rookie of the Year and led the league in rushing yards that season.