He crossed the threshold into another realm of baseball greatness on Saturday. Only 26 other major league baseball players had hit 500 home runs.

David Ortiz – now with 500 after hitting two on Saturday in a 10-4 win over the Rays – has become the 27th.

But does long-ball greatness make Ortiz one of the all-time greats? Of the other 26 members of the 500 club, 16 are also members of the Hall of Fame.

Five of the 10 non-members are not yet eligible because they haven’t been retired for five years (Ken Griffey Jr. goes on the 2016 ballot). The other five don’t deserve membership yet, according to the voters.

Will Ortiz?

Two factors may stall his membership.

One is silly: Ortiz is a designated hitter. The other is that 700-pound elephant in the room: performance-enhancing drugs.

First, the designated-hitter argument. Even though the American League implemented the DH in 1973, there are purists who believe a designated hitter is not a complete baseball player.

That argument may be what’s holding up Edgar Martinez. In an 18-year career, Martinez played 68 percent of the time as a DH. His career numbers include an impressive .312 average and .933 OPS. He hit 309 home runs.

This year was Martinez’s sixth on the Hall of Fame ballot and he was 12th in the voting, named on 27 percent of the ballots (induction requires 75 percent).

Besides being a DH, Martinez may be fighting small market (Seattle) bias, as well as minuscule postseason numbers. Although he torched the Yankees in the 1995 division series (batting .571 with two home runs, three doubles and 10 RBI), Martinez batted only .206 in his remaining 29 playoff games, with 14 RBI.

Designated hitters do make it. Frank Thomas played 57 percent of his games as a DH and was inducted in 2014. He batted .301 over his career with a .974 OPS. Like Martinez, Thomas never played in a World Series (he was injured when the White Sox won in 2005).

Of course, Thomas was a two-time MVP (when he was mostly playing first base) and hit 521 home runs.

And that brings us to Ortiz’s credentials. His career average is .284 with a .924 OPS. And he hits the ball far.

Factor in Ortiz’s postseason prowess: .295/.962 over 82 games, with 17 home runs and 60 RBI. He was named MVP of the 2004 American League Championship Series and the 2013 World Series.

Then there is Ortiz’s positive presence, from his rallying speech after the Boston Marathon bombing, to his leadership in the clubhouse and camaraderie with opponents.

Ortiz sounds like a slam dunk for the Hall. But there’s that issue of the supposedly confidential drug testing in 2003 by Major League Baseball which, at the time, was trying to determine if a drug policy was needed. Some names of violators were eventually leaked out and Ortiz’s name was among them.

While Ortiz claims he never knowingly took steroids – and he’s never failed a drug test since – the suspicion is there that some of his home runs were aided by more than muscle and a sweet swing.

All five 500-home run hitters who have not reached the Hall of Fame despite their eligibility have been tainted by at least the suspicion of PEDs. Rafael Palmeiro (569 home runs) is the only one of the five who has failed a drug test, and he’s no longer on the Hall of Fame ballot (having failed to reached the required 5 percent of the voting) to stay on the ballot.

Mark McGwire (583 homers) has admitted taking steroids. He received 10 percent of the vote in 2015.

Barry Bonds (the home run leader with 762) received 36.8 percent of the vote this year. He has long been suspected of steroid use and was in the infamous Mitchell Report on PEDs, commissioned by Major League Baseball. Bonds was the subject of a federal investigation that resulted in an obstruction of justice conviction – which has since been overturned.

Sammy Sosa (609) became a famous basher at the same time as McGwire. Although long suspected, Sosa denied using PEDs. Unlike McGwire and Bonds (who are still active in baseball, McGwire as a coach, Bonds as a consultant in spring training), Sosa has faded away. He was on 6.6 percent of the Hall of Fame ballots in 2015.

Gary Sheffield (509 home runs) received 11.7 percent this year, his first on the ballot. A crowded field of candidates may have kept Sheffield from getting more, but he was also mentioned in the Mitchell Report. Sheffield has denied ever taking steroids.

Ortiz has made denials, too. There is no reason not to believe him, except others have lied before him.

That dark cloud likely won’t keep Ortiz from the Hall of Fame. He just may not be a first-ballot selection.

Should Ortiz be inducted into the Hall of Fame? Of course. His home runs, career statistics and postseason performances make him one of the best.

He belongs with the other baseball greats.