Catalyzes the ATP-dependent phosphorylation of L-homoserine to L-homoserine phosphate. Is also able to phosphorylate the hydroxy group on gamma-carbon of L-homoserine analogs when the functional group at the alpha-position is a carboxyl, an ester, or even a hydroxymethyl group. Neither L-threonine nor L-serine are substrates of the enzyme. ATP + L-homoserine = ADP + H(+) + O-phospho-L-homoserine The catalytic efficiency is 500-fold higher with L-homoserine than with D-homoserine as substrate. Amino-acid biosynthesis; L-threonine biosynthesis; L-threonine from L-aspartate: step 4/5. Belongs to the GHMP kinase family. Homoserine kinase subfamily.