In an interview with the WSJ, Icahn signaled that Illumina "should bring Flatley back as CEO immediately." The billionaire did not disclose if he was in touch with Flatley, according to the report.

Icahn, who owns 1.4% of Illumina, is pushing for Illumina to unwind its buyout of cancer detection test maker Grail and is seeking three board seats after weeks of private discussions failed to yield results.

Icahn and Illumina did not immediately respond to Reuters' request for comment.

Icahn's campaign at Illumina could become the year's most high-profile proxy contest after Nelson Peltz ended his quest for a seat at Disney and Daniel Loeb reached an agreement with Bath & Body Works.