The Montreal-based company declined to say Friday if it will increase its C$450 million bid, which includes $78 million of debt.
"Right now we're evaluating our options," said spokeswoman Sandy Vassiadis.
After making the announcement earlier this month, chief executive Lino Saputo Jr. described the offer as a "very fair price" but hinted that it may be adjusted as "circumstances change."
The target company, Warrnambool Cheese and Butter, urged its shareholders to hold tight while it reviews the AUD$7.50 per share cash offer from Murray Goulburn Co-operative.
Warrnambool's board had unanimously backed Saputo's AUD$7 per share bid, provided it didn't receive a superior offer.
Saputo (TSX:SAP) has long searched for an opportunity to enter the Oceania market, and using it as a springboard to further service Asia's growing market, which it currently supplies from Argentina
The bid for one of Australia's largest and oldest milk processors followed more than a decade of what Warrnambool's CEO described as strategic and cordial "dialogue" between the two companies.
Faced with intensified competition and limited growth opportunities in Canada, Saputo has focused on expansion in the United States, Latin America and Oceania. Australia exports nearly half of its milk production, accounting for about 10 per cent of the global export market.
Murray Goulburn mounted an unsuccessful effort in 2009 to buy Warrnambool and currently owns about 18 per cent of its shares.
Its offer is conditional on being supported by Warrnambool's board and approval from Australia's competition authorities.
If a takeover is successful, the two Australian dairies would process up to 4.5 billion litres of milk, representing 41 per cent of the market share on milk in the country.
Also mounting a bid is Bega Cheese, which also owns about 18 per cent of Warrnambool shares and holds a seven per cent market share. Its executive chairman suggested to Australian media that it will consider raising its cash and share bid totalling AUD$370 million, or an implied price of AUD$6.30 per Warrnambool share.
Both companies are touting their Australian roots to counter their Canadian rival.
Analyst Keith Howlett of Desjardins Capital Markets describes the transaction as very small for Saputo, which has a book value of some C$2.4 billion, but significant for the two Australian bidders. Bega's book value is AUD$250 million while Murray Goulburn's is AUD$650 million.
"Our expectation is that Saputo will increase its bid for Warrnambool next week," he wrote in a report.
Howlett said as Saputo awaits an official response from Warrnambool's board on Murray Goulburn's offer, it will likely consult with advisers on whether the potential deal could face hurdles related to competition law.
Irene Nattel of RBC Capital Markets said Saputo's current offer would only increase the company's earnings by two per cent.
"Near term value (is) more strategic than economic," she wrote. "It's unclear what Saputo's next move will be, but Saputo is a disciplined acquirer."
Warrnambool's shares jumped more than four per cent to AUD$7.89, suggesting investors may be expecting higher bids.
The heated interest in Australia comes as Canada signed an EU trade deal that will double cheese exports to Canada by European dairy producers.
Saputo said it doesn't yet know what kind of impact the deal will have on its business.
"But if you look in the past, Saputo has always adapted and so that's what we will do once we have all the relevant information," said Vassiadis..
Saputo's shares gained 41 cents at C$51.57 in Friday afternoon trading on the Toronto Stock Exchange.