According to Matt Lombardo of NJ.com, the Eagles made Burton an offer that he "didn't consider serious," and it would have to be improved for him to return.
Burton reached hero status in Philadelphia when he threw a touchdown pass to quarterback Nick Foles just prior to halftime of the Eagles' 41-33 Super Bowl LII win over the New England Patriots on a play dubbed the "Philly Special."
The 26-year-old Burton is a four-year NFL veteran who is arguably coming off his best season after registering 23 receptions for 248 yards and a career-high five touchdowns in 2017.
After catching just three passes in his first two seasons, Burton broke out in 2016 with 37 grabs for 327 yards and one score.
Teammate Zach Ertz has developed into one of the NFL's elite tight ends, though, so Burton is firmly behind him on the depth chart.
The former University of Florida standout has a chance to land a sizable contract and a starting job in free agency after thriving in a supporting role.
If Burton does sign elsewhere, then Ertz will continue to be Philly's go-to tight end, and veteran Brent Celek could move back into the No. 2 spot behind him.