I currently have a multiple entry visitors visa. Can I travel to the USA and apply for an H1B visa over there?
If you can find an employer who is willing to sponsor your H1B visa and if you qualify for an H1B visa, you can get an H1B visa. However, the H1B visa has an annual quota and it opens only on Oct 1 of the given year and the applications are accepted starting Apr 1 of the given year. If you are not able to stay in the U.S. till Oct 1 on a visitors visa, you need to return to the home country, get the H1B stamped and go back to the U.S. to work on the H1B visa.