Gillett was a co-owner of the Reds between 2006 and 2010, with Tom Hicks, before the club was sold to the Fenway Sports Group, owned by John W Henry.
The four-year period saw a gradual slowdown for the club that put the Reds on the brink of administration, and also saw Rafa Benitez sacked in 2010 after six years in Merseyside.
Foster was appointed manager at Anfield during his father’s tenure as co-owner of the Reds, and recently had talks with the Derby County championship team over a possible takeover.
Foster made an offer to buy the Scottish club Hearts, which was recently relegated from the Premiership, according to a report in The Mirror.
However, Hearts president Ann Budge told BBC Scotland Hearts that he rejected a formal takeover approach to Foster.
She said: “My plan has always been to hand the club over to the fans and nothing has changed.
“Right now, the club needs stability. “
Scottish clubs voted to end the season amid the coronavirus pandemic, which means Hearts has been relegated to the championship.
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Despite Hearts’ relegation, they say they will take legal advice and insist that no club “should be unfairly penalized.”