"Initially she will be leaving with a one-month visa to perform umrah [a minor pilgrimage to Mecca] and her permanent residence there will be finalised upon reaching the kingdom of Saudi Arabia," the paper said, quoting a senior government source.
Several Bengali newspapers also reported Khaleda's possible departure for a long exile abroad.
State of emergency
Both of Zia's sons have been arrested since the interim government imposed a state of emergency on January 11 and began a crackdown on corruption.
Arafat was arrested by army-led joint forces early on Monday, following the arrest of Tareque Rahman, Zia's elder son and the political heir apparent, last month.
Government officials were not immediately available for comment.
More than 160 senior politicians have been arrested on charges of graft and abuse of power since the government imposed the state of emergency and postponed an election scheduled for January 22 in the wake of widespread violence between political factions.
Bangladeshi politics has been polarised for 10 years between Zia's Bangladesh Nationalist Party and Sheikh Hasina, the leader of the Awami League.
Hasina, who began a private visit to the US this month, has since been charged at home with murder and corruption.
In an interview in Washington on Monday, she said she would return to Bangladesh next week to fight what she called the false and fake charges against her.
Security forces dropped Arafat at Zia's home in Dhaka at early on Tuesday.
Zia has been under virtual confinement at her home for more than a week.