The surprise bid would allow Iliad, controlled by French tycoon Xavier Niel, to become Europe's sixth largest mobile operator, in a further expansion of the company's business outside France, where its low-cost mobile offers shook up the market nearly a decade ago.
Iliad has already bought a minority stake in Ireland's eir and launched in Italy in 2018.
Iliad's shares were down by 3% at 0941 GMT.
"To embark on a new adventure in Poland is a surprise for investors who would have liked to see first the benefits of the improvements in France and Italy," said Stephane Beyazian, an analyst at Main First Bank.
Iliad has been fighting cut-throat competition in France which put pressure on its shares and led Niel to tighten his grip on the company via a 1.4 billion-euro share buyback.
The group, whose stock has gained 40% since the start of the year, managed to attract new customers in France as well as securing an 8% market share in the mobile business in Italy. But the need for more spending on networks spooked investors when the company reported first-half results.
The deal, which gives Play an enterprise value of 3.5 billion euros ($4.2 billion), includes 2.2 billion euros for Play's equity, or 39 zlotys per share, a 38.8% premium to its Friday close.
The Paris-based group said it had already agreed to buy 40% of Play from two private investors, Tellerton Investments and Kenbourne Invest II, respectively controlled by Greek businessman Panos Germanos and Icelandic businessman Thor Bjorgolfsson, according to regulatory filings.
The two investors agreed to exit as the whole industry faces heavy costs upgrading to the next generation of mobile internet, or 5G.
Play followed the same playbook as Iliad in Poland, with aggressive prices in mobile that helped it to take the number one position in the market with a 29% market share of users.
"We have the same maverick mindset," Iliad's chief executive Thomas Reynaud said in a call with analysts. "Each time we have cracked the market and revolutionised the access to mobile data."
The fierce price competition hit sales revenues overall in the Polish market, which are now stabilising.
Play competes against Orange Polska, Deutsche Telekom's T-Mobile and Polkomtel [PTL.UL], a unit of Cyfrowy Polsat. It launched its broadband services this year, leaving room for expansion in this business.
Iliad said the acquisition will be earnings-accretive from year one and the offer, which starts Oct. 19 and closes Nov. 17, will be financed by debt and cash.
($1 = 3.7556 zlotys)
($1 = 0.8429 euros)
(Reporting by Geert De Clercq and Mathieu Rosemain; Additional reporting by Douglas Busvine and Anna Koper; Editing by Kim Coghill/Sherry Jacob-Phillips/Alexander Smith/Jane Merriman)
By Mathieu Rosemain, Anna Koper and Geert De Clercq