"Qatar has invited Taya to seek asylum there," said a senior Gambian official in the capital Banjulon Tuesday, where Taya is currently in temporary refuge.
He said Taya would leave for Qatar within 72 hours but did not give an exact time.
A Mauritanian diplomatic source also said Qatar had made an offer. Officials in Qatar were not immediately available for comment.
A 17-member military council led by some of Taya's closest aides seized power in the Islamic Republic on 3 August, while the president was out of the country, promising democratic elections within two years.
The putsch won widespread support in Mauritania, with jubilant residents taking to the streets to celebrate the end of Taya's 21-year authoritarian rule.
The United States, the European Union and the African Union (AU) among others have condemned the coup, but Washington and diplomats in Mauritania have said they are prepared to work with the military government if it shows it can keep its promise of polls.
Taya has urged soldiers to resist the country's new leadership and vowed to return to his home country soon, although his words were not taken seriously by residents and officials in Mauritania's capital, Nouakchott.