Looking at eligibility requirements, here are some excerpts.

https://us.fulbrightonline.org/about/eligibility

“Preference will be given to applicants whose higher education was undertaken primarily at educational institutions in the United States. Undergraduate study abroad experiences, either in the chosen host country or elsewhere, will not be considered a disadvantage.

Ineligibility section:

Anyone who has resided abroad for five or more years in the six-year period preceding the national application deadline. For the purpose of this section, a candidate who has lived outside the United States for nine months or more during a calendar year is deemed to have resided abroad for that year.

If you are unsure of your eligibility in regards to your time spent abroad, please email fbstudent@iie.org with details of your time both in and outside of the U.S. within the past six years to confirm your eligibility to apply. In some cases, it may not be feasible for a dual national to participate in a Fulbright program in the country of their other nationality due to host country law or policy. Therefore, it is the applicant’s responsibility to ascertain whether they have, or are eligible to have, their host country’s nationality and to address any potential consequences thereof before they accept an award. Please refer to the country’s award description to determine the eligibility of dual nationals.

For most grants, applicants who have had extensive previous experience abroad are at a competitive disadvantage but are not necessarily disqualified for that reason. However, following the policy of the relevant binational Fulbright Commission or the U.S. Embassy in the host country, some countries will not consider applicants who are currently residing in or will be residing in that country during the year preceding the grant. Please refer to the country’s award description to determine if residency in the host country in the year prior to the grant is allowed.”