The deal has not been announced but reports out of San Francisco indicate the trade is done and Milwaukee will be getting catcher Andrew Susac and pitching prospect Phil Bickford.
Smith, 27, has a 3.68 ERA in 22 innings for Milwaukee this season. He began the season on the disabled list with a sprained right knee and did not make his debut until early June.
The Giants have been seeking bullpen help for most of the month and reportedly had been interested in Aroldis Chapman when the team visited the New York Yankees on July 22-24.
Recently, the Giants placed left-handed Josh Osich on the disabled list with a strained right forearm. Also, left-hander Javier Lopez has a 4.00 ERA this season.
Smith has a strikeout rate of nine per nine innings and is on a team-friendly control. He is under contract through 2019 since he is an arbitration eligible player.
Last year, Smith had a 2.70 ERA with 12.9 strikeouts per nine innings.
Susac is blocked behind the plate by Buster Posey and could get more playing time if Jonathan Lucroy gets traded by Milwaukee. He has a .240 average in 87 games with the Giants over parts of the previous two seasons and was batting .273 in 58 games with Triple-A Sacramento this campaign.
Bickford has split this season with Single-A Augusta and Double-A San Jose and is 5-6 with a 2.71 ERA in 17 starts. The 21-year-old was the 18th overall pick in the 2015 draft.