Industry analysts said it was unlikely to be enough to get a deal done but it could bring Shire to the negotiating table. Several analysts have valued Shire in the mid-50s pounds per share or higher.
However, with AbbVie's shares falling 3 percent on Tuesday, the actual value of the latest offer has fallen to 50.19 pounds per share, which Panmure Gordon analyst Savvas Neophytou said was inadequate and was likely to be rejected by the Shire board.
Other analysts said Shire's board might try to seek alternative offers in order to push the price higher, although AbbVie's Gonzalez told Reuters he was not aware of any counterbidders interested in the company.
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Shire shares were down 1.3 percent at 44.70 pounds by 1120 GMT.
The silence from the Shire camp and the failure by AbbVie to land a decisive blow has prompted some hedge funds to take profits, according to people close to the situation.
Worries that the two sides may not manage to reach a deal, as happened with Pfizer-AstraZeneca, has fuelled volatility in Shire shares, which fell back sharply on Tuesday after initially rising on news of the raised bid.
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AbbVie may have some room to offer more but is likely to be constrained by a desire to maintain its investment grade credit rating.
The U.S. company is eager to buy Shire both to reduce its tax bill by moving its tax base to Britain - a tactic known as inversion - and to diversify its drug portfolio by adding Shire's specialised drugs for hyperactivity and rare diseases.
Moody's Investors Service said the latest increased offer was credit negative as it would raise AbbVie's financial leverage, with its ratio of debt to earnings before interest, taxes, depreciation and amortisation (EBITDA) expected to reach a range of 3.2 to 3.6 times, compared to 2.2 times at the end of March.