The Yankees have scored the third-most runs in the majors, which is pretty good, regardless of the circumstances.

If they could actually get some production out of the first-base spot, they’d be even better.

The position has been a disaster as both Greg Bird and Chris Carter have had brutal starts to the season.

Entering Monday’s games, the Yankees had a major-league-worst .558 OPS (Texas is next at .602) and just seven RBIs from first base.

“It’s not a concern, it’s just a fact,” general manager Brian Cashman said of the woeful state of first base. “Bird is hurt, and Carter has not hit.’’

Bird was supposed to emerge as the everyday first baseman, building on his brief success at the end of the 2015 season before he was sidelined last year by a torn labrum in his shoulder.

A strong spring ended with Bird fouling a ball off his right ankle, and he struggled before landing on the disabled list. He’s out indefinitely as the Yankees wait for him to be symptom-free before allowing him to start a rehab. He’s staying back in New York for a second straight road trip.

That leaves Carter, who led the National League in homers a year ago, hitting 41 with Milwaukee.

Since signing with the Yankees after failing to land a job as a free agent until the middle of February, Carter has been awful. In 75 plate appearances, he has just 13 hits, and just four for extra bases. There’s been just one homer — and 28 strikeouts.

Carter has been bad enough that Joe Girardi resorted to using Matt Holliday at first in a game at Yankee Stadium against Houston. Just last month, Girardi opted to not even put Holliday at first when the Yankees played in Pittsburgh for three games. Instead, Holliday was used only as a pinch-hitter.

When Holliday has been at first, he’s been able to make most plays, but doesn’t look comfortable.

He said he’s willing to grab his glove whenever called upon, but it’s clear neither side is interested in making it a regular occurrence.

The other potential option is Tyler Austin, who has been out with a fractured foot since spring training. He is due to start a rehab assignment this weekend, according to Cashman, and then is expected to be sent to Triple-A Scranton/Wilkes-Barre.

Cashman declined to speculate whether Austin would be an option when he’s healthy, and is holding out hope that either Carter starts to hit or Bird gets — and stays — healthy.

“Carter is very streaky and hasn’t gotten hot yet,” Cashman said. “Bird hasn’t been himself all season. It’s been a non-productive position, which is surprising because of the amount of personnel we had there. You’d think between the trifecta, if one faltered, another would pick it up, but that hasn’t happened yet.”