NEW DELHI—The U.S. is encouraging India to take a more-active role in Afghanistan after years of keeping New Delhi's participation limited in deference to Pakistan.

Defense Secretary Leon Panetta arrived in New Delhi on Tuesday and attended meetings with Indian Prime minister Manmohan Singh and National Security Adviser Shiv Shankar Menon. On Wednesday, he is due to meet with the defense minister, A.K. Antony, and deliver a speech on the growing U.S.-India partnership.

In...