Ryan said that the decision would be made by Ray Davis and Bob Simpson, co-chairmen of the ownership group. Davis attended the Winter Meetings for two days before leaving on Wednesday.
The Rangers have an interest in both free agents and continue to have discussions with Hamilton's agent, Mike Moye, and Greinke's agent, Casey Close. But the talks are still in the preliminary stages, according to Ryan.
"I don't think a lot of numbers have been talked about," he said.
The Rangers want to upgrade their starting pitching by adding Greinke -- whom Ryan considers the premier pitcher on the free-agent market -- to a rotation that includes Yu Darvish, Matt Harrison, Derek Holland and Alexi Ogando.
Greinke could command as much as six years for an annual salary of between $20 million and $25 million. Ryan is uneasy talking about giving pitchers long-term contracts of that magnitude, but he seems comfortable if that's what it takes to sign Greinke.
"I don't expect [Greinke] to have any arm troubles because of his history, his arm action and his effortlessness, as far as the way the ball comes out of his hand," he said. "Some guys I don't like their arm slot, their delivery or the way the ball comes out of their hand. But he's a guy, when you watch him, you feel like he's going to give you 200 innings."
Greinke went 15-5 with a 3.48 ERA and 200 strikeouts in 212 innings over 34 combined starts for the Angels and Brewers this past season. He has never been on the disabled list with an arm injury, although he missed the first month of the 2011 season with a fractured rib, an injury sustained while playing basketball. He has pitched 1,035 innings over the past five years, 14th highest among Major League pitchers in that stretch.
Greinke is also one of 17 pitchers to make at least 160 starts since the beginning of 2008, and his 3.54 ERA in the past five years ranks eighth among the 17. He is third in strikeouts per nine innings (8.75) and strikeouts per walk (3.86).
"It's hard to say you would be comfortable signing anybody for six years until you know the parameters," Ryan said. "But when I look at him, I don't have any concerns about his longevity. I look at him and feel comfortable about his durability."