A football team that is “knocking on the door” of a touchdown has the ball deep into the opponent’s territory, usually within the 20-yard line or the 10-yard line. The “door” is the end zone. “Dartmouth had been knocking on the door of a touchdown or at least a goal from the field” was cited in print in October 1924.
 
   
5 November 1923, Biddleford (ME) Daily Journal, “Biddleford High Outclassed Bridgton Academy and Won 19 to 7 at Prospect Park,” pg. 3, col. 3:
The Biddleford team showed more strength than the score indicated, as they were knocking on the Bridgton door late in the first half, and withonly a yard to go for a first down a Bridgton forward pass was intercepted.
 
26 October 1924, The Sunday Herald (Boston, MA), “Eddie Dooley is hero as Dartmouth gets 6-0 victory over Harvard,” sec. 2, pg. 1, col. 7:
Already, you know, Dartmouth had been knocking on the door of a touchdown or at least a goal from the field.
 
2 December 1934, Morning Advocate (Baton Rouge, LA), “Colgate Tactics Used To Produce Dramatic Finish,” pg. 2, col. 6:
The Bengals came right back after that and were knocking on the door.
   
5 October 1941, Boston (MA) , “Army Scales Citadel, 19-6, In Blaik Debut,” pg. 30, col. 6:
The Cadets were knocking on the door of another tally when the game ended.
 
Google News Archive
29 November 1942, (Wilmington, DE), “Middies Take Over Army, 14-0,” pg. 24, col. 3:
Just before the eriod ended, thanks to a 43-yard run by Ben Martin, Navy was knocking on the door again, reaching the Army three.
 
Google News Archive
3 November 1946, Eugene (OR)

, “Colorful Gaels Trounced 46-20” by Dick Strite, pg. 6, col. 3:
Before the opening quarter ended the Bruins, after another pass interception, were knocking on the door from the Saint 8-yard line.
     
Google Books
More Strange But True Football Stories
By Zander Hollander
New York, NY: Random House
1973
Pg. 69:
With only two minutes gone in the final period, the Rams are knocking on the door.
 
Google Books
“Braggin’ rights”
By Bill Cromartie
Atlanta, GA: Gridiron Publishers
1978
Pg. 156:
Sparked by a Pettus run of 17 and a tremendous catch by Lauder of a Wood pass, Auburn was knocking on the door with a first down at the Alabama two.
 
Google Books
The Line Formation
By Pat Flynn
St. Lucia, Queensland: University of Queensland Press
2006
Pg. 197:
The Armadillos were knocking on the door all third quarter but only came away with a field goal, and now it’s Hope’s chance to steal the ball game.
 
Google Books
Red Zone
By Tiki Barber and Ronde Barber with Paul Mantell
New York, NY: Simon & Schuster
2010
Pg. 49:
Early in the fourth quarter, they marched eighty yards on the ground, and were knocking on the door again with third and goal to go from the eight yard line.
 
Google Books
Believe:
My Faith and the Tackle That Changed My Life
By Eric LeGrand and Mike Yorkey
New York, NY: HarperCollins Publishers
2012
Pg. ?:
We fell behind 17—13, but we were knocking on the door at the 6-yard line midway through the fourth quarter.