Catalyzes the ATP-dependent phosphorylation of L-homoserine to L-homoserine phosphate. ATP + L-homoserine = ADP + H(+) + O-phospho-L-homoserine Amino-acid biosynthesis; L-threonine biosynthesis; L-threonine from L-aspartate: step 4/5. Belongs to the GHMP kinase family. Homoserine kinase subfamily.