TPG Capital and the Ontario Teachers' Pension Plan want to buy parts of Fairfax Media including its newspapers and property advertising arm, Domain.
A deal could be worth as much as A$2.5bn (Â£1.4bn).
However, loss-making Fairfax said the offer "may not optimise shareholder value".
It also said the proposal may not get off the ground "given the complexity involved in splitting the businesses".
A deal would also need approval from the federal government's Foreign Investment Review Board.
Fairfax, which also publishes The Age and the Australian Financial Review, posted a loss of A$893m last year as it struggled with plunging print advertising revenue and newspaper sales.
Last week journalists at the Sydney Morning Herald and The Age in Melbourne went on strike for a week in protest after management announced more job cuts.
However, the Domain division, which accounts for half of Fairfax's profits, is growing strongly and the company wants to spin it off and retain a majority stake.
Domain is Australia's second-biggest real estate website after one owned by Rupert Murdoch and is valued by analysts at about A$2bn.
According to Fairfax, the consortium has offered 95 cents a share for Domain, along with the metropolitan newspapers, an events business, and a start-up investing arm.
That would leave existing Fairfax shareholders with assets including newspapers in regional Australia and New Zealand, as well as stakes in the Macquarie Radio Network and Stan, a video-streaming service.
Shareholders would also be left with existing debt.
Lee Mickelburough, of Henderson Global Investors, which owns about 5% of Fairfax shares, said: "It's a troublesome structure to say that we get 95 cents for the good business and you get to keep the debt for the transition businesses. It's cheeky, the way they've structured it."
Shares in Fairfax gained 2.4% in Sydney on Monday following news of the bid, valuing the company at about A$2.5bn.

An increase led to 1% of benefits being unpaid during the year, Department for Work and Pensions figures reveal.
A year ago, a committee of MPs told the government that delayed benefits meant families were going hungry.
The government said it was providing more help to claimants to provide accurate information.
Nearly 65% of underpaid benefits - the equivalent to about £1bn a year - was the result of inaccurate information from claimants, it said.
People on Employment and Support Allowance - a benefit for those who are unable to work owing to illness or disability - were underpaid the most, with underpayments totalling £350m during the year.
The most common cause was the result of errors by claimants in recording the amount of income that they received - particularly in other benefits.
"We are committed to ensuring people receive the benefits they are entitled to. Our frontline staff discuss all the support that is available with claimants and we work closely with charities and other organisation to keep people informed of their entitlements," said a spokesman for the DWP.
"We have extensive guidance on Gov.uk and advisers available over the phone and in person to help people through the claiming process. We are also simplifying the system with the introduction of Universal Credit."
But David Samson, the welfare benefits project manager at the Turn2us charity, said that this support was not always easy to get.
"People are finding it harder and harder to access face-to-face help with completing these forms, which is why this problem goes unnoticed. This is a particular concern for people living with a mental health problem or who have learning difficulty who may need the extra support when completing them.
"Many find the forms complicated and lengthy and assessing your average income level can be especially tricky.
"We have seen a huge increase in people using our benefits calculator to work out how their income could impact on their entitlements."
The latest figures showed the biggest rise in benefit underpayments was in Pension Credit, a top-up to the state pension for low-income pensioners.
With this benefit, the DWP admitted that the biggest cause was official error.
A year ago, the work and pensions select committee said that, while many parts of the welfare system worked well, underpayments needed a higher priority.
Benefit problems "often led claimants to face difficult decisions over whether to pay their rent or provide essentials such as food, gas and electricity for their household", it concluded, with many becoming reliant on food banks as a result of underpaid benefits.
Overpayments of benefits totalled £3.3bn, the equivalent of 1.9% of benefit payments, the DWP figures show.
The owner of Australian newspapers including the Sydney Morning Herald is in takeover talks with a North American private equity consortium.
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A total of £1.7bn in benefits was not paid to those entitled to the money in 2015-16 owing to fraud and error, a new record rate of underpayment.