NEW DELHI—Santosh Pillai was wooed to work in the U.S. for his coding skills more than a decade ago and has built a good life in Cupertino, Calif. He considers it home and is awaiting approval for his green card—but is now worried his family could be forced to leave.

As President Donald Trump follows through on his campaign promises to tighten America’s borders, Mr. Pillai fears he, his wife and two children may have to return to India.

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