American International Group Inc.’s mortgage insurer, United Guaranty Corp., is seeking to raise about $700 million to $800 million as it sells shares to the public, according to people familiar with the matter.

If successful, the offering would give United Guaranty a market value of about $4 billion, said the people, who asked not to be identified because the plans are private. AIG, which is facing sustained pressure from activists to shrink, said it aims to sell as much as 19.9 percent of its stake in the initial public offering before it eventually fully exits the subsidiary.

AIG Chief Executive Officer Peter Hancock has fielded calls from activist investors Carl Icahn and John Paulson to pursue a more extensive breakup of the company. Meanwhile, Hancock has been under pressure to boost shareholder returns after posting three straight unprofitable quarters.

United Guaranty is the largest mortgage insurer by market share among its U.S. peers, according to Bloomberg Intelligence. Shares of its closest competitor, Radian Group Inc., dropped 8 percent this year amid stricter capital rules and increased competition.

AIG said in a first-quarter earnings presentation that United Guaranty has a book value of about $3.28 billion. That means the mortgage insurer would trade at about 1.2 times book value at AIG’s target valuation — a premium to Radian’s ratio of roughly 1 to 1.

United Guaranty formally filed for an IPO in March, with plans to list on the New York Stock Exchange. JPMorgan Chase & Co. and Morgan Stanley are underwriting the deal.

Representatives for AIG, JPMorgan and Morgan Stanley declined to comment.