Protein-lysine methyltransferase that selectively trimethylates residues in heat shock protein 70 (HSP70) family members. Contributes to the in vivo trimethylation of Lys residues in HSPA1 and HSPA8. In vitro methylates 'Lys-561' in HSPA1, 'Lys-564' in HSPA2, 'Lys-585' in HSPA5, 'Lys-563' in HSPA6 and 'Lys-561' in HSPA8. L-lysyl-[protein] + 3 S-adenosyl-L-methionine = 3 H(+) + N(6),N(6),N(6)-trimethyl-L-lysyl-[protein] + 3 S-adenosyl-L-homocysteine Interacts with heat shock 70 family members; at least some of these proteins are methylation substrates. Belongs to the methyltransferase superfamily. METTL21 family.