"I'm willing to get serious," said Cashman, who once again was scheduled to talk with agent Darek Braunecker on Tuesday night.
Cashman said Braunecker is setting the pace of the negotiations so the Yankees must go step for step with him.
A baseball source told ESPN.com's Jerry Crasnick the chances of Lee signing a contract before the end of the winter meetings is "doubtful."
In an effort to explore other options, Cashman said he has met with an unnamed player and might meet with two others. The Yankees have interest in Carl Crawford as a fallback option if Lee falls through.
The Texas Rangers also continued their pursuit of Lee on Tuesday. Team president Nolan Ryan said that he's more encouraged now than he was a week ago about the prospect of his club landing the pitcher.
"There doesn't seem to be anybody that's stepped out and seems to be the lead pursuer," Ryan said.
The Rangers met briefly on Tuesday for the second straight day with Braunecker, and more meetings are expected, possibly as early as Tuesday night.
Meanwhile, the Lee race welcomed a new participant Tuesday, as the Los Angeles Angels reached out to representatives for the pitcher, a source told Crasnick.
The Nationals agreed to a $126 million, seven-year contract with outfielder Jayson Werth last weekend, and there was speculation Tuesday that Washington will offer Lee a seven-year deal.
Speaking with Washington-area reporters, Nationals general manager Mike Rizzo downplayed his team's chances of signing Lee.
"I still think we're a real long shot to acquire the player," he said.
While there have been reports of possible seven-year offers for Lee, the Yankees ideally would like to limit a Lee contract to six seasons. They reportedly may go as high as $25 million per year.
While some may consider it a bluff, the Yankees have indicated that they don't want to see their budget rise above the $200-210 million mark.
"I'm a budget guy, you know that," Yankees owner Hal Steinbrenner said. "But, we have some money to spend. If we have money coming off [the books], we put most or all of it back in. That's what the fans expect. They expect us to field a good team every year. So we expect to field a good team every year and we are going to do that."
Would Steinbrenner let Cashman extend the budget?
"I'm always willing to entertain [new concepts] whenever someone comes in to my office," Steinbrenner said.
Like Cashman, manager Joe Girardi said on Tuesday he was unsure if Andy Pettitte will return or retire. Pettitte's decision likely won't come until after Lee's. For now, Lee is the Yankees' focus.
"I know what we are willing to do and how far we are willing to go," Cashman said of the Lee contract. "We would love to add Cliff Lee to the Yankee rotation. He knows that. I have a great working relationship with the agent. The player has good ties to some of the players on the team. But, at the end of the day, it is business. It is all business."
Lee and his family are very close with CC Sabathia's family from their days playing together in Cleveland.
"As a free agent, they dictate the pace," Cashman said. "We have to go the dance step. That's how it is."
Andrew Marchand covers the Yankees for ESPNNewYork.com. Follow him on Twitter Information from ESPNDallas.com's Richard Durrett was used in this report.