Congress won just 44 seats in the parliamentary election for 543 seats as the Hindu nationalist Bharatiya Janata Party (BJP) swept to power with the first majority in 30 years.
Sonia, the 67-year-old Congress party president, entrusted campaigning for the first time to her son and vice president Rahul, whose lacklustre performance failed to convince voters as Congress sought a third term in power.
"They both offered to resign but the party rejected it unanimously," member of parliament Amarinder Singh told reporters after a meeting of the party's top decision-making body in New Delhi.
Party spokesman Janardan Dwivedi said the Congress Working Committee had passed a unanimous resolution "to let Sonia Gandhi take whatever corrective steps are needed" to revitalise the party.
The Nehru-Gandhi dynasty has provided three prime ministers, including independent India's first, but analysts are increasingly questioning if it is the vote-winner it used to be.
"Economic growth and social mo