According to NFL Network's Ian Rapoport, the Vikes offered Cousins a three-year deal worth $28 million per season.
Rapoport added that while nothing is set in stone, Cousins is "likely" to sign with Minnesota if his free-agency visit goes well.
ESPN's Adam Schefter echoed that Cousins' signing with the Vikings is "likely." He also tweeted a statement from Cousins' agent, Mike McCartney, who said: "No final decisions have been made. The plan is to visit Minnesota first. That's all we can say right now."
Cousins will have dinner with Vikings coaches and decision-makers Wednesday night before visiting the team facility Thursday, per Rapoport.
Free agents can officially sign with teams beginning at 4 p.m. ET on Wednesday.
The 29-year-old Cousins spent the first six seasons of his career with the Washington Redskins, but his tenure essentially came to an end when Washington agreed to acquire quarterback Alex Smith in a trade with the Kansas City Chiefs.
Cousins is a one-time Pro Bowler who has topped 4,000 passing yards and thrown for at least 25 touchdowns in each of the past three seasons.
In 2017, he completed 64.3 percent of his passes for 4,093 yards, 27 touchdowns and 13 interceptions along with four rushing scores.
The Vikings mainly utilized Keenum under center last season, but Sam Bradford and Teddy Bridgewater also saw minimal action.
All three could sign elsewhere in free agency.
If Cousins signs with Minnesota, he will join a team that reached the NFC Championship Game and boasted the NFL's top defense in yards allowed last season.