It said the offer was equivalent to £49 a share - a 60% premium to its closing price on 27 March, before Takeda made its interest known.
All the proposals consisted of a mix of cash and Takeda shares. Representatives for Takeda and Shire declined to comment.
The offer values Shire at approximately £46bn and shareholders will be entitled to any dividend prior to the takeover, which will see shareholders own 50% of the enlarged Takeda.
The companies now have until May 8 under an extended deadline to thrash out the finer points of the deal and while Takeda kicks the tyres at Shire before its board gives final approval.
Under British takeover rules, Takeda had until 5 p.m. (1600 GMT) on Wednesday to make a firm offer or walk away from Shire. It would also be the third biggest acquisition of a UK-traded company in history.
Last week, Botox maker Allergan said it was considering making a counter-offer for Shire, raising the prospect of a bidding war, but then confirmed it would not make a bid.
Takeda's shares slid nearly 6 percent in early Tokyo trade on Wednesday as investors fretted over its ability to finance the cash and stock deal. It specializes in medicines for rare diseases, as well as the ADHD drug Adderall.
It announced last week a sale of its oncology business to unlisted French drugmaker Servier for $2.4 billion. That is much smaller than the value of the fourth bid for Shire that Takeda disclosed last week. Hence why the overall bid and cash has had to be revised higher (£21.75/share represents a 36% improvement on the Takeda's first proposal), to sweeten the deal and try to compensate for some of the share price decline.
Shire traces its roots back to 1986, when it began as a seller of calcium supplements to treat osteoporosis, operating from an office above a shop in Hampshire, southern England.
Takeda bought United States biotech Ariad Pharma for $4.7billion a year ago, although many market commentators felt this was overpriced.