If reached, the deal would likely be for four years, the sources said.
An early, unconfirmed report said Hamilton wants $175 million for seven years. Most teams would prefer a shorter deal; Hamilton is a recovering drug addict with a history of injuries.
His offensive performance, however, is elite. Hamilton batted .285 last season with a career-high 43 homers and 128 RBI. He is a lifetime .304 hitter with a .913 on-base/slugging percentage.
It's still possible that another team could beat the Rangers' offer.
Mariners officials met with Hamilton in Nashville on Sunday, according to major league sources. Other unidentified clubs also are believed to have met with Hamilton.
The Milwaukee Brewers are on the periphery of the Hamilton sweepstakes, according to those same sources. Johnny Narron, the Brewers’ hitting coach, previously was Hamilton’s accountability partner and an assistant hitting coach with the Rangers.
Hamilton, a Christian, has said in the past that God will help him make his free-agent decision, an indication that he might not go to the highest bidder or best team.
Hamilton’s wife, Katie, told Sports Illustrated in June that the family planned to donate a significant portion of Josh’s next contract to charities.
Pro Athletes Outreach, a Christian organization that ministers to athletes, coaches and their families, is based in Issaquah, Wash., a suburb of Seattle.
“No, we don't really care about the money so much for us, but we have huge plans for this money and, no, it's not strictly for our bank account,” Katie Hamilton told SI. “It is for a hurting world.