The Boston Globe was the first to report Burkhead staying in New England.
The Patriots have already lost Danny Amendola, Malcolm Butler and Dion Lewis to free agency.
Burkhead, who entered the NFL as a sixth-round draft choice of the Cincinnati Bengals in 2013 out of Nebraska, had signed a one-year contract with the New England Patriots on March 14, 2017. The one-year deal had a maximum value of $3.15 million.
Burkhead played in 10 regular-season games (three starts) for New England, as injuries to his ribs and knee limited him. He finished with 264 yards on 64 carries with 5 touchdowns, adding 30 catches for 254 yards and 3 touchdowns.
The Patriots were deep at running back, and Burkhead ultimately settled into the No. 3 spot on the depth chart when healthy. He played 17.2 percent of the offensive snaps in the regular season, as Dion Lewis and James White were ahead of him.
The 5-foot-10, 210-pound Burkhead also played on the punt return and kickoff return units, which added to his value on the 46-man roster over fellow running back Mike Gillislee.
Burkhead turns 28 on July 2.
ESPN's Mike Reiss contributed to this report.