Visa transfer to new passport

I found my passport with a valid US Visa after having lost it initially. I now have a new passport as well. Can the existing visa on my previous passport be transferred to the new passport?

Have you reported the old passport - and/or the visa - as lost to the authorities (local, of the country of issuance, etc)?

You cannot "transfer" a visa from a passport to another. Normally you can travel with both passports and show to the US border officers the one with the stamped visa, as long as the visa is still valid (not expired and not reported as lost to the US authorities).

Don't forget,what makes a visa unique are the marginal annotations inscribed on the visa and the lower parts of any visa.
It contains, full name, DOB, passport number, nationality and the works. The minute you surgically "lift" it to another passport, it becomes invalid.

As advised, use both passports (new and old), provided visa remains valid.