The newspaper, citing multiple baseball industry sources, said the 35-year-old Australian's deal is worth a total of $15 million and is pending a physical.
He would replace former all-star Jim Johnson, who was traded to Oakland Dec. 2 for Jemile Weeks and a minor league player to be named later.
Balfour is coming off a career-high 38 saves for the A's last season. He compiled a 1-3 record with a 2.59 ERA in 65 relief appearances.
He took over as Oakland's full-time closer in 2012, helping the team to back-to-back American League West titles.
In his 10-year career, Balfour has logged a 28-17 record with a 3.27 ERA and 514 strikeouts over 463 games with Minnesota, Milwaukee, Tampa Bay and Oakland.