Near rhyme: Where the rhyme repeats, the sound is close but not exactly the same (given the varieties of dialect, this is probably typical of even most "perfect" rhymes in English).

                     Will't please you sit and look at her? I said
                   "Fra Pandolf" by design, for never read
                     Strangers like you that pictured countenance,
                     The depth and passion of its earnest glance,
                     But to myself they turned (since none puts by
                     The curtain I have drawn for you, but I)...

                            from "My Last Duchess," Robert Browning                        
                            (1842)

    The first two lines of the clip are perfect rhymes. The second pair are near. The third pair are nearer.

    Friend, your fugue taxes the finger
      Learning it once, who would lose it?
    Yet all the while a misgiving will linger,
      Truth's golden o'er us although we refuse it-
    Nature, thro' cobwebs we string her.

                   from "Memorabilia," Robert Browning (1896)

          "Linger" and "string her" are a near rhyme. "Lose it" and "refuse it" might be in some dialects.

                      So we'll go no more a roving
                      So late into the night,
                      Though the heart be still as loving,
                      And the moon be still as bright....

                               from "So We'll go no more a roving,"
                               Lord Byron, 1817

"Roving" and "loving," in 1817, may have been exact rhyme but we don't have any direct means of finding out.   Now, outside of Liverpool, they're near, though some may argue strictly for their being consonant rhyme.

            She'd the brooch I had bought
                         And the necklace and sash on,
                  And her heart, as I thought,
                            Was alive to my passion;
          And she'd done up her hair in the style that the Empress
             had brought into fashion.

                            from "Atalanta in Camden Town," Lewis
                            Carroll (1869)

"Sash on" and "passion" are a fine near rhyme.

    Cold-blooded, smooth-faced, placid miscreant!
       Dabbling its sleek young hands in Erin's gore,
      And thus for wider carnage taught to pant,
       Transferr'd to gorge upon a sister shore...

                   from "Don Juan," Byron (1824)
   
"Miscreant" and "taught to pant" are a good near rhyme.

Mom says that Amos owes me an apology. And a lot of punitive damages.