Shire's support for the higher offer from US-based AbbVie is dependent on a number of conditions being met.
Founded in the UK in 1986, Shire is best known for its attention deficit disorder drugs. It is listed on the London Stock Exchange, has its tax base in Dublin and sells most of its products in North America.
AbbVie's pursuit of Shire comes just weeks after UK firm AstraZeneca fought off takeover interest worth £69 billion from US drugs giant Pfizer.
AbbVie, which has until Friday to confirm an offer for Shire, plans to move its domicile to the UK in a move that would cut its corporate tax rate.
The Illinois-based company was formed in January 2013 after it was spun off from Abbott Laboratories. It employs around 25,000 people worldwide and markets medicines in more than 170 countries.
Shire held talks with AbbVie over the weekend following the receipt of proposals in which its shareholders would hold 25% of the combined business. The proposal values Shire at £53.20 a share, equivalent to £31.35 billion.
AbbVie first approached Shire at the start of May, with the third proposal by the end of that month valuing the company at 4626p a share, or £27.2 billion.
Shire's products enable people with life-altering conditions to lead better lives.
It provides treatments in areas such as rare diseases and neuroscience and is developing products in other therapeutic areas. Basingstoke-based Shire's best selling product is hyperactivity drug Vyvanse, which achieved revenues of 1.2 billion US dollars (£703 million) last year.
The pharmaceuticals sector has seen a raft of deal-making in recent months, with US companies looking to Europe as a way to reduce their tax liabilities.