Speaking at a news conference Saturday at the Arizona Diamondbacks’ spring training ballpark, Selig said last year’s positive drug test for All-Star game MVP Melky Cabrera and allegations players received banned substances from a now-closed Florida anti-aging clinic helped lead him to seek stiffer penalties as quickly as possible.

He declined to give specifics, saying MLB Executive Vice President Rob Manfred and players’ union head Michael Weiner will meet.

Weiner said Monday that some players have expressed support for tougher penalties. Selig said he was encouraged by Weiner’s comments.

“The players have been discussing whether changes in the penalties are warranted since the offseason,” Weiner said Saturday during a telephone interview. “As I’ve said throughout spring training, there’s a variety of player views on this subject. In fact, during the offseason we suggested to the commissioner’s office the possibility of differential penalties, namely advanced penalties for certain intentional violations but reduced penalties for negligent violations.

“That format was not of interest to MLB at that time. We look forward to ongoing negotiations over the drug program, but any change in the penalties would be a 2014 issue. It would be unfair to change the drug-testing rules now that the 2013 program has begun to be implemented.”

MLB and the union started urine testing with an anonymous survey in 2003 and added penalties in 2004, when a first offense resulted in counseling. A 10-day suspension for a first offense was instituted for 2005, and the current discipline structure has been in place since the 2006 season: 50 games for an initial PEDs infraction, 100 games for a second and a lifetime ban for a third. No player has reached the third level.

The initial penalty for a stimulants offense is follow-up testing, with a 25-game penalty for a second violation, 80 games for a third and the discipline for additional offenses to be determined by the commissioner under a “just cause” standard.

Selig wants a tougher penalty for first-time offenders.

“There’s no question about that,” he said.

Twelve players were given 10-day suspensions in 2005. Thirty suspensions have been announced from 2006 on, including just two 100-game bans — to pitcher Guillermo Mota and catcher Eliezer Alfonzo. The penalty for Alfonzo was cut to 48 games because of procedural issues similar to the ones that led an arbitrator last year to overturn Ryan Braun’s positive test for elevated testosterone before a suspension was announced.

Suspensions for positive urine samples announced in 2012 increased to eight, when Cabrera, Bartolo Colon and Yasmani Grandal all tested positive for testosterone.

“We’ve made meaningful adjustments to our testing, and the time has come to make meaningful adjustments to our penalties,” Selig said. “There is no question that there have been enough events that say to me the program is good, but apparently the penalties haven’t deterred some people.”

Players and management added spring training blood testing for human growth hormone last year and in January announced a deal expanding it to the regular season. Also in January, they said the World Anti-Doping Agency laboratory in Laval, Quebec, will keep records of each player, including his baseline ratio of testosterone to epitestosterone. The lab will conduct Carbon Isotope Ratio Mass Spectrometry (IRMS) tests of any urine specimens that “vary materially.”

MLB was the first major sports league in North America to test for HGH.

Selig said those who have violated the anti-drug rules are “a very small percentage” of players.

“A great majority really, really have been terrific,” Selig said, “and I give the players’ association a lot of credit. We had lots of problems two decades ago, 10 years ago, but I’m confident that Michael and Rob will sit down, because I feel very strongly about this.”

Cabrera, who was leading the NL in hitting for the San Francisco Giants, was suspended for 50 games last year. After asking for a rules change that prevented him from winning the NL batting title, he signed a $16 million, two-year contract with Toronto during the offseason.

Selig would not comment on the now-defunct Biogenesis of America clinic in Coral Gables, Fla., other than to say it is the subject of a “very thorough investigation” by MLB.

The facility was alleged in media reports to have provided performance-enhancing substances to several players, including Cabrera, Alex Rodriguez and Nelson Cruz. The players have denied they obtained banned drugs from the clinic. MLB has been trying to obtain purported records of the clinic posted online by The Miami New Times, which initially revealed the allegations.

“The program is working fine,” Selig said, “but I’ve come to the conclusion the more I’ve thought about this that obviously there are some people, small in number, who need to be given a tougher lesson.”

In the year ending with the 2012 World Series, there were seven positives for performance-enhancing substances and 11 for stimulants among 3,955 urine tests and 1,181 blood tests, according to a report issued in November by baseball’s independent program administrator, Dr. Jeffrey M. Anderson.

“We’re way ahead of what anybody could have thought, but my father used to tell me life is nothing but a series of adjustments,” Selig said, “and this is an adjustment that you have to make based on what you see.”

Joe Torre and Tony La Russa, retired managers who work for Major League Baseball, both voiced support for the tougher penalties at Selig’s news conference.

Torre, the former New York Yankees and Los Angeles Dodgers skipper who will manage the United States in the World Baseball Classic, said it is important to remove questions fans may have about whether players are clean.

“Until we can gain the total respect back from fans and have them trust us again, we’ve got work to do,” he said.

La Russa, longtime manager of the St. Louis Cardinals, said although the punishment already is severe, it apparently isn’t enough.

“Just make that risk so punishing that we can eliminate this,” he said.

Selig said stiffer penalties are in the best interest of baseball.

“Anybody who will be dismayed by this announcement is living in a world that I don’t understand,” he said, “and in my own feeling frankly doesn’t exist.”