Anwar was convicted and sentenced to five years in prison in March on charges of sodomising a young former male aide, but is free pending a final appeal set for October 28-29.
Anwar, 67, has dismissed the case as fabricated by the government to ruin his political career.
The political aide, who asked not to be named, declined further comment but said that a media report on Monday on the asylum offers was correct.
The Star newspaper quoted Anwar saying he had received several offers by "foreign leaders" to go overseas but would not accept them.
"I met up with a few foreign leaders who advised me to go overseas to escape a possible prison sentence. At my age, they say they cannot imagine seeing me behind bars," Anwar was quoted as saying.
The story did not specify which countries Anwar was speaking of.
Sodomy is punishable by up to 20 years' jail in the Muslim-majority nation.
A lower court acquitted Anwar in 2012, but an appeals court controversially overturned that verdict in March.
Anwar, a former top leader in Malaysia's ruling United Malays National Organisation (UMNO), had earlier spent six years in jail on separate sodomy and corruption charges after a power struggle in the late 1990s that saw him expelled from the party.
The charges triggered massive protests against the UMNO-led regime.
UMNO has governed since independence in 1957, but has seen its support slump over corruption, authoritarian rule and divisive racial policies.
After his release, Anwar joined the fractious opposition, leading it to unprecedented electoral gains.
Election Results for Lok Sabha Election 2019 will be out on May 23. Get the latest election news and live updates on ndtv.com/elections. Catch all the action on NDTV Live. Like us on Facebook or follow us on Twitter and Instagram for news updates from each of the 543 parliamentary seats for the election 2019