The Yankees and Rangers are both awaiting a decision from Lee, the 2008 American League Cy Young winner and the top pitcher on the free-agent market. No other clubs have been reported as serious contenders to land the 32-year-old left-hander, although a few teams do have interest.
"For somebody of that stature, it would certainly behoove him to be a Yankee, which would probably be for the rest of his career," Steinbrenner said during a telephone interview Friday with The Associated Press. "I think that would be a great move for him but, of course I'm prejudiced."
New York reportedly offered Lee a six-year deal Wednesday worth from $137.5 million to $140 million, then increased it to seven years the following day after the Red Sox landed outfielder Carl Crawford on a deal of that length.
"Everything is progressing," Steinbrenner said. "We made the offer. It's a very good offer, and we certainly hope he takes it. It's all up to him and his agent. We'll see what happens."
Steinbrenner said he isn't concerned that Lee would be 39 in the final season of the contract, citing the success of Andy Pettitte -- who, as a free agent, is still weighing retirement. At 38 years of age in 2010, Pettitte went 11-3 with a 3.28 ERA, although he did miss two months because of a groin strain.
"Looking at how well Andy pitched up until this year and so forth, and he's a lefty, the same kind of pitcher as Andy, I don't really see a problem," Steinbrenner said. "I think Cliff's the kind of guy that can get it done and be effective for a long time. He's a great pitcher."
Greenberg visited Lee in Arkansas on Thursday and presented what he said were multiple offers. The Rangers obtained Lee from Seattle in July, and Lee led Texas to its first World Series appearance.
"The information we gave him yesterday is a lot to digest," Greenberg said. "We want them to take as much time as they need to come to a fully-informed decision that they are comfortable with."
Lee's agent, Darek Braunecker, did not respond to messages.
Southpaws might be of particular interest for the Yankees -- and the other clubs in the American League East that will face Boston on a regular basis. Boston's two big acquisitions this offseason, Adrian Gonzalez and Crawford, are both left-handed hitters.
Right-hander Phil Hughes went 18-8 with a 4.19 ERA this year and wound up as the Yankees' No. 2 starter behind Sabathia. A.J. Burnett remains in the fold and fellow righty Ivan Nova is another candidate for the rotation, especially if Pettitte retires.
"Most of the season last season, he was fantastic -- he just got tired out at the end," Steinbrenner said of Hughes. "In the second full season as a starter, he's going to be that much better. We've got some good young pitchers coming, too. Everybody thinks Nova is the real deal. And with the pitching we're going to have and the bullpen, and with the hitting we're going to have, as we always do, I would hope that Cliff would want to join the party."