By Liz Hoffman

June 26, 2015 2:42 p.m. ET

Some AOL Inc. shareholders want more than the $50 a share Verizon Communications Inc. paid to take the iconic Web company private earlier this week.

Investors owning at least about 4 million shares, or 5% of AOL, didn’t sell their stock to Verizon and instead told the company they plan to seek a higher price in court, through a legal process known as appraisal, according to people familiar with the matter.

The dissenting funds include Brigade Capital Management LP and Verition Fund Management LLC, some of the people said. Brigade filed a notice with the Delaware Court of Chancery that it plans to seek an appraisal for about 1 million shares.

The funds have a few months to abandon the appraisal plans and accept the deal price.

The $4.4 billion deal closed Tuesday. About 60% of AOL shares were tendered into Verizon’s offer, according to a securities filing, a low number for an uncontested merger. The stock had consistently traded above the deal price in recent weeks, likely on hedge funds buying shares to preserve their appraisal rights. An AOL spokesman declined to comment.

Appraisals, in which investors ask a judge to award a higher price for their shares, have gone from a little-used remedy for slighted shareholders to a bona fide investing strategy. A record 40 such cases were brought last year, and another 28 have been filed already this year representing claims with a face value of about $1.8 billion, according to a review of court filings.

Retailer PetSmart Inc., fruit grower Dole Food Co. and jewelry-store chain Zale Corp. all have multimillion-dollar appraisal actions pending from recent buyouts.

The AOL-Verizon deal is an unusual target, though. Most appraisals have focused on buyouts by private-equity firms, particularly those involving management, where they say asymmetrical information and big insider stakes can skew deals in favor of a single, preferred bidder. Arm's-length sales to corporate buyers are less often challenged.

The investors don’t yet have to say how much they think their AOL shares are worth. Analyses by AOL’s bankers at Allen & Co. suggested a range of $42.92 to $58.82, using a variety of metrics.

Write to Liz Hoffman at liz.hoffman@wsj.com