However, in regards to a potential trade, nothing is imminent, according to sources.
The Jets are in a stalemate with free agent Ryan Fitzpatrick and are nowhere close to an agreement, sources told Schefter.
The Browns are expected to part ways with Johnny Manziel this offseason and with the No. 2 overall pick in the draft, are reportedly thinking about selecting a quarterback at that spot.
Kaepernick's agents requested permission from the 49ers on Feb. 26 to seek a trade for their client.
The 49ers quarterback is recovering from surgeries to his left shoulder, right thumb and left knee, and on April 1 his $11.9 million base salary for 2016 becomes guaranteed. There also is a seemingly growing distrust between player and organization after he was benched midway through the season and after a 2-6 start.
Kaepernick, 28, chose to have private doctors perform his procedures, rather than team doctors.
At the scouting combine last month, 49ers general manager Trent Baalke said Kaepernick will be on the team's roster on April 1, when his base salary becomes fully guaranteed.
Kaepernick is under contract through the 2020 season as part of the six-year, $114 million contract he signed in 2014.
That contract included a $12.3 million signing bonus, $61 million guaranteed, and an average annual salary of $19 million. In 2016, Kaepernick has a cap hit of $15.8 million, while his dead money value is $7.3 million.
If he stays with the 49ers In 2016, Kaepernick will be playing for his third head coach since he joined the team in 2011, as former Eagles coach Chip Kelly was hired to replace the fired Jim Tomsula. Kaepernick was drafted in the second round of the 2011 draft, when Jim Harbaugh was the team's coach.
His career record as a starter is 27-20 with 56 touchdown passes and 26 interceptions, although he is coming off the worst season of his career, having been benched after eight games in favor of Blaine Gabbert.
The 49ers on Thursday signed quarterback Thad Lewis to a one-year contract. He becomes the fourth quarterback on the team's roster.
ESPN 49ers reporter Paul Gutierrez contributed to this report.