A source told ESPN NFL Insider Adam Caplan that the Bears' offer to Hill is a three-year deal worth a maximum of $7.5 million, with $3.25 million guaranteed.
The Saints tendered Hill at the lowest level for a restricted free agent. They have five days to match Chicago's offer or lose Hill without receiving any compensation.
The Saints could let Hill walk, because they signed tight end Coby Fleener to a five-year, $36 million contract and re-signed veteran backup Michael Hoomanawanui. New Orleans also entered the week with a league-low $1.3 million in salary-cap space, and that was before their deal with linebacker James Laurinaitis had been processed.
Hill, who is 6-foot-5, 250 pounds, recently visited Bears headquarters at Halas Hall outside of Chicago. He has 36 receptions for 340 yards and eight touchdowns over three seasons in New Orleans. He caught a career-high five touchdowns in 2014.
Hill is also a core special-teamer.
"This Josh Hill is another player that I love. I love," Saints coach Sean Payton said last March. "Josh Hill is a player that we value a lot. He's not on anyone's fantasy first three rounds, I promise you. But he's a real good tight end. He's versatile in the running game, in the receiving game, as a special-teams player."
Bears general manager Ryan Pace was the Saints' director of player personnel during Hill's first two years in New Orleans.
Hill is a former undrafted free agent out of Idaho State.
The Bears clearly have a need at tight end after trading Martellus Bennett to the New England Patriots. Newly re-signed Zach Miller is the favorite to start in 2016, but Chicago wants to strengthen its depth at the position. The Bears also brought back Rob Housler on a one-year deal and traded for Khari Lee last summer.
ESPN Saints reporter Mike Triplett contributed to this report.