McInnes is an experienced manager in Scotland having started by taking St Johnstone to the top flight in 2009.
After a brief stint at Bristol City he then had seven years at Aberdeen where they never finished lower than fourth and won the League Cup in 2014.
McInnes departed Pittodrie in 2021 and took charge of Kilmarnock the following January, leading them to promotion back to the top flight and keeping them up on their return.
His spell at Rugby Park peaked last season when he guided the Ayrshire side to fourth place and European football.
Hearts themselves have had a turbulent season which started with their worst-ever beginning to a campaign and cost Naismith his job.
Critchley steered the club away from the bottom of the table but failed to get them into the top six and was dismissed, while there are changes off the pitch.
Brighton owner Tony Bloom is set to invest nearly £10m in the club, which has use of his Jamestown Analytics data company for recruitment.
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