Perrigo unanimously rejected Mylan's earlier $205 per share offer on Tuesday, saying the bid substantially undervalued the company.
Perrigo's shares fell 2 percent to $197.58, while Mylan's stock rose 2.2 percent to $75.36 on Friday morning.
"Mylan has today begun a legally-binding process under the Irish Takeover Rules to commence its offer for Perrigo ...," Executive Chairman Robert Coury said in a statement on Friday.
Mylan said its offer is fully financed, cash confirmed and not conditional on due diligence.
The cash portion will be financed by a new bridge credit facility arranged by Goldman Sachs, the company said.
Read MorePerrigo board unanimously rejects Mylan proposal
Mylan's pursuit of Perrigo, a major producer of over-the-counter medicines, is seen as an apparent attempt to stave off interest from Teva.
Teva said in an emailed statement on Friday that the company remained fully committed to its Mylan offer.
A Perrigo spokesman declined to comment.
The three-way chase is further evidence that the appetite for healthcare acquisitions continues unabated with M&A in the industry hitting $193.9 billion this year, double 2014 levels.
Teva shares were up 1.5 percent at $64.20.
CORRECTION: An earlier version misstated the value of the proposed offer.