The offer was submitted to AT&T Wireless' board of directors over the weekend, according to the source, who spoke on condition of anonymity.
The source did not say how much Cingular was offering but said it was entirely in cash.
A message was left for AT&T Wireless, based in Redmond, Wash., but it was not immediately returned.
USA Today reported Tuesday, citing unidentified executives at two companies involved in the deal, that Cingular's offer includes a premium over AT&T Wireless' closing share price of $9.99 on Friday. Cingular, jointly owned by SBC Communications and BellSouth, is currently the nation's second-largest wireless carrier behind Verizon Communications. AT&T Wireless is the third-largest.