Languages differ in their realization of the subject argument: non-null-subject languages, like English, require subjects to be phonologically overt; rather, null-subject languages, like Greek, allow the subject to be overt or null. This cross-linguistic difference can lead to the transfer of grammatical properties across languages during bilingual language acquisition. The direction of crosslinguistic transfer is known to be affected by structural overlap and linguistic dominance. This exploratory study was conducted to establish whether structural overlap or linguistic dominance affect the production patterns of Greek children who acquire English at preschool. If structural overlap determines the direction of transfer, then children will overuse overt subjects in Greek; if dominance determines the direction of transfer, then children will overuse null subjects in English. Two groups of Greek children (between 3;6-4;5 and 4;6-5;8) attending a monolingual English immersion programme in Greece participated in a novel oral elicitation task that tested their use of null and overt subjects in both languages. Both groups produced significantly more null and fewer overt subjects in English than an English monolingual control group, but the same number of null and overt subjects in Greek as a Greek monolingual control group. This finding suggests that the preschoolers, who start learning English at 3;0 years, experience crosslinguistic transfer from Greek, their dominant language, to English -- thus highlighting the role that dominance plays in determining the direction of crosslinguistic transfer among successive bilingual children.