Ends of human chromosomes consist of the six nucleotide repeat d[pTTAGGG]n known as telomeric DNA, which protects chromosomes. We have previously shown that the DHX36 gene product, G4 Resolvase 1 (G4R1), binds parallel G-quadruplex (G4) DNA with an unusually tight apparent Kd. Recent work associates G4R1 with the telomerase holoenzyme, which may allow it to access telomeric G4-DNA. Here we show that G4R1 can tightly bind telomeric G4-DNA, and in the context of the telomeric sequence, we determine length, sequence, and structural requirements sufficient for tight G4R1 telomeric binding. Specifically, G4R1 binds telomeric DNA in the K+-induced "3+1" G4-topology with an apparent Kd = 10 ± 1.9 pM, a value similar as previously found for binding to unimolecular parallel G4-DNA. G4R1 binds to the Na+-induced "2+2" basket G4-structure formed by the same DNA sequence with an apparent Kd = 71 ± 2.2 pM. While the minimal G4-structure is not sufficient for G4R1 binding, a 5' G4-structure with a 3' unstructured tail containing a guanine flanked by adenine(s) is sufficient for maximal binding. Mutations directed to disrupt G4-structure similarly disrupt G4R1 binding; secondary mutations that restore G4-structure also restore G4R1 binding. We present a model showing that a replication fork disrupting a T-loop could create a 5' quadruplex with an opened 3'tail structure that is recognized by G4R1.