The Niners are expected to give Ford a multiyear deal in the range of five years and $87.5 million, the source said.
San Francisco will send a 2020 second-round pick to Kansas City, a source confirmed. The deal can't become official until the new league year begins Wednesday.
"I feel appreciated," Ford told Yahoo Sports. "Me knowing they believe in me ... talking to the GM, talking to the owner, talking to the head coach ... they couldn't wait [for me to get there]."
The Chiefs had placed the franchise tag on Ford last week, with the intention of trying to trade him, as reported by ESPN's Adam Schefter.
Kansas City has hired Steve Spagnuolo as defensive coordinator and is changing its base defensive system from a 3-4 to a 4-3, which would have necessitated a position change for Ford and Justin Houston, who was released by Kansas City.
Last season, Ford finally became the player the Chiefs had envisioned when they drafted him out of Auburn in the first round in 2014. He started all 16 games for the first time in his career and had a career-high 13 sacks. Ford, who will turn 28 on March 19, also was selected to play in the Pro Bowl for the first time.
Ford ended his best season with his least productive game. He had no sacks or tackles in the Chiefs' AFC Championship Game home loss to the New England Patriots. He also was penalized for lining up offside on a play that resulted in an interception in the final minute of the fourth quarter.
The interception could have allowed the Chiefs, who had a four-point lead at the time, to run out the clock. Instead, the Patriots continued their drive to a go-ahead touchdown and eventually won in overtime.
For the 49ers, adding a premium edge rusher -- or two -- has long been at the top of their offseason shopping list. During the 2018 season, coach Kyle Shanahan compared the importance of an edge rusher to the defense to that of a quarterback on offense. It's a sentiment that San Francisco general manager John Lynch shared at the NFL scouting combine last month.
"You have to have a guy who can close out games, close out halves," Lynch said then. "When you have them, they can be a difference-maker. I think we're also looking for powerful players. ... They're going to be playing with a quick trigger and getting after it. But those are guys that can get to the quarterback and finish him."
San Francisco has been searching for a player on the outside fitting that description since it parted ways with Aldon Smith in 2014. Last year, interior lineman DeForest Buckner became the first Niner to reach double-digit sacks since Smith in 2012, but Shanahan and Lynch have been searching for the outside complement to help a team that has lost 11 one-possession games over the past two seasons turn that tide.
The 49ers finished 2018 tied for 22nd in the NFL in sacks with 37, but aside from Buckner, they did not have another player with more than 5.5 sacks.
San Francisco also holds the No. 2 overall pick in this year's draft, which could still allow the 49ers to find a pass-rushing tag-team partner for Ford.
MMQB first reported on the trade agreement.
ESPN's Adam Teicher and Nick Wagoner contributed to this report.