According to the New York Post, some of the city’s best-known eateries are lucky the Health Department is starting to hand out letter grades next week — instead of last month — because thousands would have ended up with a bottom-rung "C" plastered in their front windows.

Officials estimate that about 6,000 of the city’s 24,000 eateries had enough violation points in June to have earned the lowest mark on a three-letter rating scale devised by the city.

The "C" restaurants would have ranged from the Lion, a sizzling new spot in Greenwich Village, to the venerable Gallagher’s steakhouse in Midtown, to the century-old Katz’s deli emporium on the Lower East Side.

Even Radio City Music Hall’s snack bar made the "C" list.

The Health Department plans to award "A" grades to restaurants that accumulate no more than 13 violation points; "B" to those with 14 to 27 points; and "C" for 28 or more points.

Restaurant owners and managers contacted by The Post who would have faced a "C" last month were surprisingly supportive of the grading system.

"It’s for the sake of public health — I’m perfectly OK with that," said Jake Dell, son of the owner of Katz’s deli, which accumulated 47 points on its record for such infractions as evidence of roaches and mice, as well as bad plumbing.

Like every restaurateur contacted, he said the conditions cited by inspectors have since been corrected. A reinspection July 6 brought Katz’s score down to 23 — in the "B" range.