The offer of 313p per share is a 63% premium to Aldermore's float price and would value the firm at over £1 billion.
The two companies are in preliminary discussions about the details of the possible offer, Aldermore said, and in accordance with City code FirstRand has until 1700 GMT on November 10 to announce its firm intention to make such an offer.
Earlier, Aldermore confirmed that it had received a 313 pence per share acquisition offer from FirstRand.
Shares in Aldermore were trading 18% higher at 302.80p on the back of the news.
The group - which lends to small and medium-sized businesses, home-owners and landlords - saw mortgage lending increase by 9% to £6.2 billion. "This announcement has been made without the consent of FirstRand and the board will keep shareholders updated as appropriate".
"FirstRand has been assessing opportunities to build a sustainable long-term deposit franchise to fund its strategy to grow and diversify the revenues of its current United Kingdom business".
Listed challenger bank Aldermore Group has revealed it has been approached about a possible takeover.