Denver can't officially agree to a deal with the free agent until 4 p.m. on Wednesday - when the NFL's free agency period officially opens - but a source told 9News Denver that Keenum and the Broncos will have a short-term deal in place.
The Vikings, who were subsequently blown out at Philadelphia in the conference championship, chose to let all three of their QBs - Keenum, Bradford and Teddy Bridgewater - test free agency.
Keenum finished the season with 3,547 yards passing to go with 22 touchdowns and seven interceptions.
The Vikings have no quarterbacks under contract for 2018 but have been linked with both Kirk Cousins and Drew Brees, who has yet to be re-signed by the Saints.
Six years ago, Elway landed Peyton Manning, the biggest free agent prize in National Football League history, following his release by the Colts and a series of neck fusion surgeries. The Vikings are considered the favorite for Cousins with the New York Jets and Arizona also in the mix.
It didn't look like Keenum would become much more than a journeyman backup after stints with the St. Louis Rams, back to Houston, then to the Los Angeles Rams, before landing in Minnesota last season.
Manning had a spectacular closing chapter in Denver, leading the Broncos to the playoffs in all four of his seasons, reaching two Super Bowls and winning one. "That is what we are working on".
Denver were also reported to be in the hunt for Cousins having struggled through three options under centre last season but general manager John Elway has made his move for Keenum.
Elway had said early in the offseason, "For us to have a chance to get better, we have to get better at that position". We didn't play well there a year ago. However, that wasn't good enough for him to get drafted. I feel like we can get right back in the thick of the things rather quickly.