Deoxyribonucleoside kinase that has a broad specificity phosphorylating thymidine, 2'-deoxyriboadenosine, 2'-deoxyribocytidine and 2'-deoxyriboguanosine. Specificity is higher for pyrimidine nucleosides. Several anti-viral and anti-cancer nucleoside analogs are also efficiently phosphorylated. a 2'-deoxyribonucleoside + ATP = a 2'-deoxyribonucleoside 5'-phosphate + ADP + H(+) Subject to feedback inhibition by dTTP. Monomer. Belongs to the DCK/DGK family.