The newspaper claims that Irvine's name was volunteered by Steven Naismith, the current Hearts striker.
And he is said to be a contender for the Edinburgh side's vacant manager and sporting director jobs.
Irvine was last employed as assistant to another of his former Everton colleagues, David Moyes, at West Ham United.
But he previously the number two at Norwich, where he reunited with Naismith in July 2016 after the pair had spent season together at Everton earlier in the decade.
Irvine, who hails from Glasgow, has also managed in his own right at Preston, Sheffield Wednesday and West Bromwich Albion.
Norwich are reported to have paid Everton around £8 million for Naismith, but released the 33-year-old at the end of his contract earlier this year.
Everton and Norwich fans - would Irvine be a good appointment for Hearts?
Have something to tell us about this article?