Avery, 30, dealt with injuries all season, including a sports hernia that required surgery in October and a persistent groin injury. He appeared in just six games in 2014, catching 15 passes for 176 yards and losing one fumble. Avery was signed by Kansas City in 2013.
In 22 games for the Chiefs, he had 55 receptions for 772 yards and two touchdowns.
Jenkins was drafted 30th overall by the San Francisco 49ers in 2012. In three NFL seasons, the 25-year-old has recorded just 17 receptions for 223 yards.
After being acquired by Kansas City through a trade in Aug. 2013, he appeared in 25 games during his time with the Chiefs, with much of his playing time coming because of Avery's injuries last season.
Avery signed a three-year, $8.55 million contract with Kansas City in 2013, and was slated to make $3.4 million in salary and bonuses this coming year. Jenkins would have earned nearly $1.33 million. The moves will save Kansas City almost $5 million in cap space.
The Chiefs have struggled offensively in recent years, finishing no higher than 21st in the league in total offensive yards over the last four seasons. Over the same period, the team has been in the bottom half of the league in total offensive points, including last place in 2012.
The Chiefs finished 9-7 in 2014.