The club is believed to have been offered a multi-million pound loan by Leeds City Council, reports BBC Sport.
Yorkshire currently lease the ground from the Leeds Cricket, Football and Athletic company.
But a buy-out is seen as the best way of ensuring Headingley's future as an international cricket venue after the England and Wales Cricket Board decided not to stage an Ashes Test there during the summer.
International cricket is thought to boost the county's economy by £4.25million a year.
The addition of Durham's Riverside headquarters and the Rose Bowl in Southampton to the list of international venues means Tests may be awarded on a rota system in future years.
