Buddhism in ancient Arabic regions
Wikipedia's Ashoka article includes this map:



Its western border is the Seleucid empire which appears to be where modern-day Iran is (note that Iran I think might be ethnically Persian rather than Arabian).

It also includes modern-day countries like Pakistan and Afghanistan (which are Islamic but not exactly Arabic).

This map is labelled "Buddhist proselytism at the time of king Ashoka (260–218 BCE)":



It shows Buddhist proselytism reaching (beyond the Maurya empire) to Alexandria (which is in modern-day Egypt), and Antioch (modern Turkey).

This may not be "expansion of Buddhism" though, it's more like "envoys from the Buddhist empire".