Several other U.S. companies are using mergers to reincorporate in countries with lower tax rates. These moves are raising concerns among U.S. lawmakers since they can cost the federal government billions in tax revenue.
AbbVie executives also said the product portfolios of the two companies complement each other, and the combination would immediately give Shire a broader geographic reach.
Shire, which is known for its rare-disease drugs, has so far rejected three overtures from AbbVie. Earlier this week, Shire CEO Flemming Ornskov told analysts that the company should remain independent and that it expects to more than double its 2013 annual product sales to $10 billion by 2020.
Shire said Wednesday that AbbVie's offer "fundamentally undervalued" the company and its prospects, and its board has already unanimously rejected it.
The British drugmaker also has said its board had concerns about AbbVie's interest in making the move for tax purposes.
Earlier this month, U.S. medical device maker Medtronic Inc. said that it had agreed to buy Ireland-based competitor Covidien for $42.9 billion in cash and stock. The combined company would have its executive offices in Ireland, which has a 12.5 percent corporate income tax rate.
Walgreen Co., the largest U.S. drugstore chain, also is considering an option to complete a takeover of Swiss health and beauty retailer Alliance Boots, a move that could involve another overseas reincorporation.
U.S. drugmaker Pfizer Inc. also recently tried unsuccessfully to acquire U.K.-based pharmaceutical company Astra-Zeneca.
AbbVie was spun off from Abbott Laboratories at the start of last year. Its products include branded prescription drugs like the blockbuster anti-inflammatory drug Humira, which brought in more than $2.6 billion in revenue during the first quarter. The company also has about 10 products in the final phase of clinical trials before seeking regulatory approval.
"We're pursuing this transaction from a position of strength," Chairman and CEO Richard Gonzalez told analysts on Wednesday.
AbbVie has until July 18 to either announce a firm offer or confirm that it won't make one under United Kingdom takeover laws. The U.S. company said in a statement that "there can be no certainty that any firm offer will be made." It noted that the companies are no longer talking.
AbbVie's stock climbed 2.5 percent, or $1.33, to $54.92 in afternoon trading, while U.S.-traded shares of Shire rose about 3.2 percent, or $7.08, to $230.35. Broader trading indexes, in contrast, were nearly flat.