Jalna - a well-known Australian Greek yoghurt brand - has been for sale through PwC.
A2 Milk, whose market cap stands at $4.26 billion, would be a large undertaking for Saputo, which bought the financially stressed Australian dairy co-op Murray Goulburn for A$1.3b ($1.39b) in 2018.
Shares in A2 Milk have been under downward pressure since 2020, reflecting a more constrained earnings outlook for the company.
International food group Nestlé has, from time to time, been touted as a possible buyer of a2 Milk.
A2 Milk's net profit plunged 79.1 per cent to $80.7 million in the June 2021 year due to the prolonged impact of the pandemic and a rapidly changing China infant nutrition market, prompting the company to review its growth strategy.
New chief executive David Bortolussi has been charged with carving out a new future for a2 Milk - one that will involve product innovation and slimmer margins.
A2 Milk is now targeting sales revenue of $2b over the next five years or more, compared with $1.2b over the past financial year.
The company last year said its sales margins would probably be in the "teens" in the medium term - well down from the 20s and 30s seen over the last few years.
Today's share price is a far cry from the record high of $21.51 reached in July 2020, before a far more modest earnings story emerged.
Rimer did not lend much weight to the speculation.
"There is no strong strategic case for Saputo to own a2 Milk," he told the Herald.
"And it would be a large transaction for them," he said.
"It's a very different target compared with what they have bought historically, but some people are clearly responding to it (the speculation).
"We have been here several times before and nothing has materialised."
Harbour Asset has a stake in a2 Milk.
Infant formula is not an easy place to be right now, especially since the birth rate in the biggest infant nutrition market in the world - China - is slowing.
Last June, UK-based Reckitt Benckiser Group sold its infant formula and child nutrition business in China to Primavera Capital Group for US$2.2 billion ($3.28 billion).
Late last year, the Dutch dairy cooperative Royal FrieslandCampina was reported by financial news wires as wanting to sell its Friso infant nutrition brand.
Buyout firms Bain Capital and Carlyle Group are among the suitors for Friso, Bloomberg reported.
Baring Private Equity Asia Ltd. and Shijiazhuang Junlebao Dairy may also consider bidding for the asset, which could be valued at about $1b to $2b, the agency said.