Cabinet spokesperson Bandula Gunawardena on Tuesday said Mr Rajapaksa is "not hiding" and "may eventually consider returning to Sri Lanka but there is no definite political or other stance on this".
His statement comes amid reports of the former president eventually moving to the United Arab Emirates.
Mr Rajapaksa, 73, fled the cash-strapped island nation along with his wife and two bodyguards after anti-government protests demanding his resignation intensified with demonstrators storming into the presidential palace in Colombo.
He left for the Maldives on 13 July, before landing in Singapore for a “private visit” the next day.
The former president emailed in his resignation from Singapore on 14 July as debt-ridden Sri Lanka battled its worst economic crisis since independence in 1948.
The Rajapaksa government has been blamed for the crippling financial crisis that has left the nation's 22 million people short of fuel and food.
Since April, thousands of people have taken to the streets in protest, which has eventually forced the entire government to step down.
While Mr Rajapaksa was initially issued a short-term visit pass of 14 days upon arrival in Singapore, it was extended for another two weeks by the government. He will be now allowed to stay in the southeast Asian nation till 11 August.
The Singapore government said Mr Rajapaksa had not been granted asylum, but he was in the country on a private visit.
Sri Lanka's supreme court has barred the president's brothers - former prime minister Mahinda Rajapaksa and former finance minister Basil Rajapaksa - from leaving the country till 2 August.
The apex court issued an order from preventing the former ministers from leaving the country until 28 July, which was extended following a petition filed by non-profit Transparency International Sri Lanka (TISL) and three others.
Meanwhile, a rights group urged the Singapore government to immediately arrest the former president for committing war crimes during Sri Lanka’s civil war and for breaching the Geneva Conventions in 2009 when he was the defence minister.