Chords Notes

Dm7 F Gm A7

Dm7:
D F A C

F:
F A C

Gm:
G Bb D

A7:
A C# E G

A Bb C C# D F G

Dm7 Analysis

D F A C in Dmin Scale
D root
F minor 3rd
A 5th
C minor 7th

D F A C in Cmaj Scale
C root
D major 2nd
F 4th
A major 6th

A C D F in Amin
A root
C minor 3rd
D 4th
F minor 6th

The analysis of Dm7 reduced the potential keys to 3, C, Am and Dm where C and Am are interchangeable, so basically C major or Dmin.

F Analysis

F A C

Diatonically in:

Cmaj/Amin
Fmaj
Dmin

Gm Analysis

G Bb D

Gmin
Dmin
Fmaj
Bb maj

A7 Analysis

A C# E G

Dmaj
(Dmin)

Result

All 4 Chords have Dmin in common

Taking Dmin as the tonal center now:

D E F G A Bb C

So it is a i–III–iv–V7 progression.

Further refining.
♭III, because F is a minor 3rd of the D minor.
V⁷, because A major triad's Bb is not naturally in D minor, so we "burrow" the raised 7th, from D harmonic minor .
It stays in D minor, but interprets the last chord as a harmonic one to resolve the the tonic stronger.

Another way to to look at it is