The Melbourne-based company also lowered its full-year earnings estimates and said it will sell its Asian pathology business after receiving a number of approaches.
The bids come less than four years since the country's second-biggest private hospital operator went public. During that time, the firm has issued at least two profit warnings as Australians increasingly opted for public health services, pushing its shares below their initial public offering price.
Healthscope said both $3 billion-plus offers, from Canada's Brookfield Asset Management and jointly from local private equity firm BGH Capital and 14 percent shareholder AustralianSuper, were conditional and undervalued the firm.
"The Directors have carefully considered each proposal and concluded that neither proposal adequately reflects the long-term value of Healthscope, nor its underlying assets nor future potential," said Chairman Paula Dwyer.
Healthscope shares fell as much as 7 percent at market open, but recovered somewhat to 3.2 percent lower in afternoon trade. The stock has risen 21 percent since BGH-AustralianSuper bid $3.1 billion (2.3 billion pounds)
for the hospital operator on April 26.
Brookfield this month offered $3.3 billion with terms that essentially prevented AustralianSuper from voting against it.
Spokesmen for both Brookfield and the BGH-AustralianSuper consortium declined to comment.
"In our view the Board will require a less conditional ... bid before it will open the books," J.P. Morgan analysts said in a research note to clients.
Healthscope said it plans a strategic review of its hospital property portfolio and would look at the merits of a sale-and-leaseback transaction to unlock value for shareholders.
"While this is not without risk given the challenging operating conditions, with multiple bids and three interested parties, we suspect it will prove successful."
However, the value of the property would then be considered as debt, J.P. Morgan analysts said.
"We are also cautious a sale and lease back will materially increase the fixed cost leverage altering the risk profile of what is traditionally considered a defensive business," the analysts said.
Healthscope also lowered its forecast range for fiscal 2018 core earnings from hospital operations to A$340 million to A$345 million, versus A$359.4 million in 2017. It previously indicated 2018 earnings would be broadly similar to 2017.
The firm has been suffering from a decline in patients since local media reported about private health insurers withholding payouts to policyholders, prompting more patients to opt for public healthcare.
On Tuesday, Healthscope also said it targeted core earnings growth of at least 10 percent from hospital operations in 2019.
(Reporting by Paulina Duran; Additional reporting by Ambar Warrick and Byron Kaye; Editing by Muralikumar Anantharaman and Christopher Cushing)
By Paulina Duran