SEATTLE — Clayton Beeter cannot escape it.

Yankees fans know his name not because of who he is, but because of a person he is not.

Beeter, the pitching prospect who was exchanged for Joey Gallo at last year’s trade deadline, wants to be more than that.

“I know every time when my name gets mentioned, so does Joey Gallo,” Beeter said this weekend at the All-Star Futures Game. “It’s cool, but at the same time, I want people to know who I am, too.”

Beeter is a 24-year-old righty who is making a strong case to be a major league starting pitcher.

The Texas native had been used with the Dodgers primarily in short bursts, never pitching more than four innings in a game in a year and a half with the organization.

The Dodgers were wary about his injury history — Beeter required a pair of surgeries while at Texas Tech, including Tommy John — and the Yankees loosened the reins.

“They’re all-in on [me as a] starter right now,” Beeter said. “That’s what I’m told, at least.”

After the trade, Beeter began to stretch out at Double-A Somerset, where he impressed last year and took off this year.

The former second-round pick pitched to a 2.08 ERA in 12 starts with Somerset this season, earning both a trip to the Futures Game and a promotion to Triple-A Scranton/Wilkes-Barre.

Beeter has been OK in two starts at the higher level and recorded two outs in relief Saturday in the Futures Game.

Beeter’s fastball touched 96 mph, and a sharp slider struck out Brewers prospect Jackson Chourio.

Those are his two primary offerings, but he has been trying to throw a developing changeup, too.

The Yankees’ starting depth has been tested during a first half in which 10 different pitchers have begun a game.

Pitchers who are projected as Triple-A depth, including Jhony Brito and Randy Vasquez, have been called upon to help.

Beeter, now one step away from The Bronx, has put himself in position to be a dark-horse candidate for spot starts.

In some organizations, young starters can break in without much pressure.

Beeter, who mentioned Matt Krook and Greg Weissert as friends to lean upon in Scranton, knows there is not much time for a learning curve if he is promoted.

“It’s probably different with the Yankees just because they’re always contending. You’ve got to go up and perform,” Beeter said. “But it’s exciting to see everybody going up, and it’s encouraging, too.”

It is possible Beeter becomes a late-season bullpen option, though it seems unlikely this year because of the club’s reliever depth and willingness to see if Beeter can last as a starter.

The early returns — as demonstrated by Beeter joining top prospect Spencer Jones for the festivities at T-Mobile Park — have been promising, which might help put Gallo a bit more in the rearview mirror.

“The whole season from the start was just [about] going deep and staying healthy, see how I responded,” Beeter said. “It’s been great so far.”