The New York Daily News’ Peter Botte and Bill Madden report that Willie Randolph’s chances of becoming the next Mets manager are improving by the day.

The Mets will move forward and conduct a second round of interviews with their three finalists for their managerial opening today after wild-card entry Jim Leyland dropped out of the running.

Leyland spoke with GM Omar Minaya and respectfully withdrew from consideration – leaving a three-horse race, with all signs pointing to Yankees bench coach Willie Randolph as the slight front-runner. Texas hitting instructor Rudy Jaramillo and Dodgers minor-league coordinator Terry Collins, whom the Mets also hold in high regard, also will be at Shea today for interviews.

“We’re most likely moving forward with these three,” Minaya said on a conference call with reporters. “I called (Leyland) this morning … and he thanked me for considering him, but the job description I presented to him was not what he was looking for right now.”

Minaya, who added that he’d like to have a manager in place by “the end of the week or early next week,” indicated that Leyland “took a pass” when he described the job as “baseball-plus.”

Minaya explained that description as including work in the community as well as several offseason evaluation trips, a commitment he wasn’t sure Leyland was prepared to give.

“The most important thing is the baseball side, but it’s fair to say I have other expectations for the manager,” Minaya said. “It goes beyond what happens during the regular season. It’s going to be a 12-month-a-year job.”

Leyland, 59, who won a World Series with Florida in 1997, confirmed last night that he remains under consideration for the managerial vacancy in Philadelphia.

“I’d like that job and I hope I’m offered it,” said Leyland, who interviewed Monday with Phils GM Ed Wade. “We also had a great conversation with the Mets. Omar was very candid and I have to respect that. He’s a real sharp guy and I know he’s going to do well there. In my opinion, he’s got it all figured out.

“But he’s looking for someone both for now and down the road. In the end, our agreement was that this was not a good fit for each party.”

Reached by telephone last night, Randolph had no such trepidation over the demands of the job and expressed excitement over advancing to the next phase. The longtime Yankee player and coach also said he hasn’t been stung by criticism he’s received from Met fans on talk radio in recent days, saying “that comes with the territory.”

“I’m a New Yorker, I earned my stripes here, I live here year-round and I can’t wait for (today) to come,” Randolph said. “I believe I can handle whatever the job entails and I’m looking forward to getting another chance to speak with them.”

Jaramillo, who still could be offered the Mets’ batting coach job if he isn’t named their manager, and Collins, who previously has skippered teams in Anaheim and Houston, also will receive the chance to further impress the Mets brass.

Minaya also confirmed that former Mets manager Bobby Valentine never was a serious candidate. Valentine’s agent had E-mailed Minaya late last week to inform him that Chiba Lotte, the Japanese team Valentine currently is managing, would’ve declined the Mets permission to speak with h