It was reported last week that the White Sox have become "increasingly willing" to move Jeff Samardzija, who is owed $9.8 million this season and will become a free agent this winter.
The Giants are looking to add a starting pitcher and would prefer to add an elite one to start a potential second playoff game behind Madison Bumgarner in October.
Other teams are still in play for Tyler Clippard, including the Nationals, but Jon Heyman of CBS Sports reports that the Yankees are out on the reliever.
Murphy, 33, is hitting .299/.347/.443 in 224 plate appearances. He is owed $6 million this season and has a $500,000 buyout on his $7 million team option for 2016.