Zain's board is expected to meet soon to discuss Bharti's offer, local daily al-Rai said in an unsourced report.
Officials at both firms were not immediately available for comment.
Bharti has resumed its hunt for emerging market acquisitions after its planned $24 billion merger with South Africa's MTN failed in September, the second time a proposed deal between the two had collapsed in just over a year.
In October, Akhil Gupta, deputy group CEO at the mobile operator's parent, said Bharti would look at buying a stake in Zain if there was an opportunity.
Ownership of Zain and its assets in Africa have been the subject of much speculation since August last year when talk of a potential sale emerged.
Zain, the third-largest telecoms operator in the Arab world, in October halted talks to sell the African assets to appease potential buyers of a 46 percent stake in the parent company, Zain Group.
A consortium of Asian investors has been trying to buy a stake from Kuwaiti family conglomerate Kharafi Group for 2 dinars per share, or about $13.7 billion.
ZAIN SHARES STRONG
Shares in Zain have risen 23 percent since February 4 and rose 3.9 percent on Thursday, ahead of the Middle East weekend.
On Monday, Zain said in a statement on the Kuwaiti bourse website that it had not received any offers for the sale of its Zain Africa assets.
Three days later, Zain appointed Nabil bin Salama as the firm's new chief executive, replacing Saad al-Barrak.
Al-Barrak, who was seen as the driving force behind the company's growth into 23 countries across Africa and the Middle East, resigned from Zain earlier this month amid uncertainty about the fate of the sale of the parent company stake.
Shares in Bharti closed on Friday at 314.4 rupees, valuing it at about $25.8 billion.
Dubai-based Shuaa Capital upgraded Zain to a buy rating. With all the news flow converging toward a potential sale of the African unit, we upgrade our recommendation on Zain from Sell to Buy, it said in a statement.
In another unsourced report, Kuwaiti al-Watan newspaper said that the firm's board is due to meet on Sunday to discuss the Bharti offer.
Last month, Bharti agreed to buy 70 percent of Bangladesh's Warid Telecom for an initial investment of $300 million. It also set up a new unit to drive its foreign expansion, focused on opportunities in emerging markets where it can replicate its low-price, high-volume model.
Bharti's home mobile market is facing margin pressures from intense competition and price wars, resulting in lower tariffs and shrinking profits.
(Reporting by Eman Goma in Kuwait and Devidutta Tripathy in New Delhi; Writing by Lincoln Feast; Editing by Louise Ireland)