You are permitted the same gift tax annual exclusion ($14,000 in 2016) as a US citizen. The consequences of a gift in excess of that amount depends on whether your are a US domiciliary for federal estate and gift tax purposes. This test is somewhat different that the test for resident or nonresident alien for gift tax purposes. For gift tax purposes, an individual is a resident if that person is domiciled in the U.S. at the time of the gift.
If you are domiciliary, you have the same estate and gift tax exemption ($5.45 million in 2016) as a citizen, and your gift in excess of $14,000 will count against that exemption. If you are not a domiciliary, your gift in excess of $14,000 will be subject to federal gift tax.
Cash held in a US bank account is considered tangible property for purposes of the federal estate and gift tax rules.
Please follow these links for more information.
https://www2.deloitte.com/content/dam/Deloitte/us/Documents/Tax/us-tax-us-estate-and-gift-tax-rules-...
https://www.clarkskatoff.com/practice-areas/estate-planning/estate-and-gift-tax-chart-for-non-us-per...
