Previous Chinese-led bids for the estate were blocked by the government, which cited national security concerns.
The Rinehart offer had hoped to appease those government concerns.
The new counter-bid has four of the country's wealthiest cattle families offering A$386m (??240m; $293m).
Kidman had already accepted the joint $A365 million offer by Ms Rinehart and Chinese developer Shanghai CRED Real Estate.
However, the company's board is required to consider any higher offers.
The Rinehart bid also is still pending approval from the country's foreign investment board.
The new offer was filed late on Sunday from the BBHO consortium, comprising the Brinkworth, Buntine, Harris and Oldfield families, and would not require such approval.
The sale of the Kidman estate, which comprises farms the size of South Korea, has highlighted the concerns over the sale of Australian assets to Chinese investors.
The stockman who built an Australian cattle empire
101,411
sqkm of overall territory
77,300 sqkm to be sold
1% of Australia to change hands
15,000 tonnes of beef carcass shipped globally per year
The landholding - with its 10 cattle ranches, a bull breeding stud and a feed lot covering 101,411 sq km (39,155 sq miles) in four states - encompasses about 1.3% of the Australian continent.
A successful bid by BBHO would create one of the world's biggest cattle businesses, with more than 500,000 head of cattle.
A spokeswoman for Ms Rinehart said the two bids were, in fact, priced "about the same" because the Rinehart bid excludes two of the Kidman properties.
These are close to a government weapons-testing range at Woomera and therefore deemed to be too defence-sensitive to go into partly foreign ownership.
A new all-Australian bid for the Kidman estate, the country's largest cattle farm, has topped an offer led by the country's richest woman Gina Rinehart together with Chinese investors.