The Marlboro owner is offering 150p per share for the London-listed target, the companies said on Friday.
The bid represents a 10% increase above a price Vectura had already agreed with private equity giant Carlyle in May.
It also represents an 11% premium on Vectura’s closing price on Thursday.
Vectura shares surged by as much as 13% in early trading on Friday after the new offer, amid speculation that further bidding could still take place.
We are delighted to welcome Vectura’s great people and management team to join us in the next stage of our mission to improve people’s lives through innovative developments in science and technology
Jacek Olczak, Philip Morris International
Directors at the inhaler specialist said they have now withdrawn their backing for Carlyle’s bid and will adjourn a shareholder meeting convened for Monday.
PMI, which also owns brands including Chesterfield and L&M, has focused its investment into its IQOS heated-tobacco devices in recent years and has also targeted strong growth away from nicotine.
It has been busy with acquisitions, announcing an 820 million US dollar (£595 million) move earlier this month to buy Fertin Pharma, which makes nicotine chewing gum and oral pain relief drugs.
PMI said it intends the acquired business to continue as an “autonomous” unit.
Chief executive Jacek Olczak said: “We are delighted to welcome Vectura’s great people and management team to join us in the next stage of our mission to improve people’s lives through innovative developments in science and technology.”
Vectura chairman Bruno Angelici said: “The acquisition by PMI recognises Vectura’s differentiated technologies, skills and expertise in inhalation and the continued development and supply of important medicines to patients.
“We recognise the material increase in the price offered to shareholders under the acquisition when compared with the Carlyle offer and have accordingly recommended the acquisition to shareholders.
“The acquisition will provide our people with the opportunity to form the backbone of an autonomous inhaled therapeutic business unit of PMI, helping develop products to improve patients’ lives and address unmet medical needs.”