The person spoke to The Associated Press on condition of anonymity Friday because the offer was not announced.
Girardi's contract expires at the end of the month. Yankees general manager Brian Cashman met with Girardi on Monday and with the manager's agent, Steve Mandell, on Wednesday.
Girardi replaced Joe Torre after the 2007 season and was given a $7.8 million, three-year contract. He is completing a $9 million, three-year deal.
He has led the Yankees to a 564-408 record and a World Series title. Crippled by injuries, New York had its poorest season since 1992, missing the playoffs for the second time in 19 years and finishing tied for third in the AL East at 85-77.
Cashman said Tuesday "we're going to give him a real good reason to stay."