The Knicks have convinced at least one faction they are extremely serious about trading for or buying a late first-round pick with the NBA Draft less than a week away.

When shooting guard P.J. Hairston agreed to work out for Knicks president Phil Jackson on June 5, it was the strongest indication they are setting their sights on more than just buying a second-rounder next Thursday night.

Hairston, the controversial former North Carolina standout who was expelled from school last December, is slated to go between the 20th and 30th picks. He has worked out for about a dozen teams, mostly clubs in the lottery.

“The interest level is very, very high,’’ Hairston’s agent, Juan Morrow, told The Post. “You never know what Phil Jackson has up his sleeve.’’

Morrow admitted even if the Knicks can’t make a deal — and there are enough teams with multiple first-round picks looking to trade, such as the Thunder and Suns — it was an honor for Hairston. Jackson took control of much of the workout, which came before new coach Derek Fisher was hired.

“Any kid growing up watching Phil Jackson’s teams, when he wants you to work out for him, you do it,’’ Morrow said.

Jackson loves big guards and Hairston fills the bill at a solid 6-foot-5, 240 pounds. It may be worth giving up on combo guard Iman Shumpert, whose rookie contract expires after this season and who could be due a bigger payday.

After his North Carolina fiasco, in which he was caught driving cars that apparently had been rented for him, Hairston latched on in the D-League for the Texas Legends in January and drove up his stock.

“He’s NBA-ready,’’ Morrow said.

One NBA general manager gave The Post this scouting report: “Two things stick out for him: his body, one of the stronger bodies of anybody in the draft for a wing position like Mitch Richmond. He’s extremely strong, a bully-type scorer. He can shoot the NBA 3 and beyond without a lot of effort because he’s so strong. He had some huge games in D-League. Downside to him, though, is as high as his highs are, his lows are just as low. He really struggled in some games. Consistency was an issue for him.’’

Hairston averaged 21.8 points in 26 games and shot 36 percent from 3-point land. In his D-League debut, he had 22 points and 6 steals. His North Carolina mistakes — he also got pulled over for speeding — seem only a passing issue because his interviews at the Chicago pre-draft camp were impressive, as was his 37-inch vertical leap.

“His character doesn’t help him, but I don’t know if it hurts him,’’ the GM said. “It may be more immaturity than bad character.”

Multiple teams have extra first-rounders. Oklahoma City is shopping the 21st and 29th and already made an offer for Shumpert at the trade deadline. Suns GM Ryan McDonough admitted this week he’s looking to trade one of his three first-round picks — the 14th, 18th or 27th. (Suns owner Robert Sarver nearly traded for Shumpert two years ago.)

The Bulls have the 16th and 19th picks and are amenable to a trade. The Sixers have two first-rounders and five second-rounders and will look to give them away for cash. The Jazz (5, 23) and Bobcats (9, 24) also have two first-rounders.

“You can see why the Knicks would have interest in [Hairston],’’ the GM said. “He’s already made a little transition.’’