The signings are contingent on whether "all goes well with COVID-19 testing and physicals" during their visits with the Pats.
Leggett, 25, was a fifth-round draft pick by the New York Jets in 2017. He's appeared in 15 NFL games (all in 2018), registering 14 catches for 114 yards and a touchdown.
He spent much of his first season on injured reserve, and the Jets cut him before the 2019 season. The Buccaneers added him to their practice squad last season but waived him this July.
In New England, he would battle with players like Devin Asiasi and Dalton Keene for playing time at tight end.
The 28-year-old Kilgo was a sixth-round pick by the Denver Broncos in the 2015 NFL draft. He's appeared in 29 total games with the Broncos (2015-16) and Tennessee Titans (2018), posting 16 tackles and a quarterback hit.
He spent time on New England's practice squad in the 2016 and 2017 seasons and also had stints with the Jacksonville Jaguars, Houston Texans and Detroit Lions.
He would compete for a depth role with the Patriots behind players like Lawrence Guy, Deatrich Wise Jr. and Beau Allen on the defensive line in the team's 3-4 scheme.