In initial talks with the Yankees, sources said Cano asked for a 10-year deal in excess of $300 million.
"I don't feel this organization is ready to do something like that," Steinbrenner said Tuesday on "The Michael Kay Show" on ESPN New York 98.7 FM. "No, I do not.
"We'll see. I know that is a number that is out there now. We'll see if he gets it, how much he wants to be in New York. But, again, I can promise it is going to be a very, very solid offer that we do make because we are going to try. We are going to try the best we can to keep him."
While he made it clear that the Yankees hope to sign Cano, Steinbrenner could foresee starting next spring with Cano in another team's training camp.
"I'm optimistic and I know that we are going to make him a very, very good offer," Steinbrenner said. "Is anybody an absolute must-to-sign? No. And that's nothing against Robby. [It's] just not reasonable to assume that about anybody. We are going to do what we can. We've certainly conveyed to Robby we want him back and we want him to be a career Yankee. We'll just have to see what transpires here."
Cano, who turns 31 on Oct. 22, hit .314 with 27 homers and 107 RBIs this season.
"There is no doubt he is a marquee player," Steinbrenner said.
In 2007, the Yankees signed Rodriguez to a 10-year, $275 million contract, which included a potential $30 million in incentives. The Yankees are waiting to find out the decision on Rodriguez's ongoing arbitration hearing over his 211-game suspension for violating the joint drug agreement and the collective bargaining agreement.
If Rodriguez's suspension is upheld for 2014, the Yankees will save at least $25 million -- and, more likely, $31 million -- off next year's payroll. Rodriguez's 2014 base salary is $25 million. If he hits six more home runs, he will pick up another $6 million for tying Willie Mays at 660 on the all-time home run list.
Steinbrenner's people said Tuesday the team's "serious goal" is to have the payroll fall beneath the $189 million threshold so the Yankees' luxury-tax rate is significantly reduced. If A-Rod is suspended, a source told ESPNNewYork.com the Yankees could re-sign Cano and possibly someone like Atlanta Braves catcher Brian McCann. If A-Rod's money is still on the books, it would be virtually impossible to sign Cano and another big-money free agent and stay under $189 million.
Steinbrenner said fielding a championship team will ultimately trump the luxury tax when making final decisions.