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The rumor was first reported by theflyonthewall.com.
Noted investor Doug Kass also commented on Twitter that he was "hearing" Icahn might have taken a position in Whole Foods.
Business Insider left a message with Icahn, but at the time of publishing had not heard back.
Citing sources familiar with the situation, Yahoo Finance's Kevin Chupka reported that Icahn has "absolutely no interest in Whole Foods whatsoever."
Following Chupka's report, shares of Whole Foods fell, and had pared gains to about 2.5%.
Tuesday's quick pop in shares of Whole Foods comes amid a terrible year for the organic grocer, which has been one of the worst performing members of the S&P 500 this year.
Including Tuesday's advance, shares of Whole Foods were down more than 31% year to date.
And following each of the company's three quarterly reports this year, the stock has sold off sharply.
This chart from Google Finance shows Tuesday's action in Whole Foods shares.
Google Finance