The Championship side have told candidates the post is only for the rest of the season — as Hoyle hopes to do a deal with a US group.
Huddersfield sit 23rd in the second tier after just seven wins from 30 matches.
Fotheringham was dismissed after just five months in the job following a 2-2 draw with Blackpool earlier this month – a game in which they conceded a 90th-minute equaliser.
And the managerless Terriers were then beaten 1-0 by fellow relegation candidates Wigan on Saturday as their troubles worsened at the foot of the table.
Fotheringham’s sacking came after he launched an astonishing rant against his players in the wake of the Blackpool draw.
He said: “We are absolutely gutted and devastated in there. There has been a lot of angry, harsh words that have been said and rightly so.
“Because we got ourselves in a situation where we [were] in pole position and surrendered three points [which] was unacceptable. I am lost for words.
“There is a real collective anger in there and a lot of harsh words said collectively as a group and we have let the club down there.
“We started really well and there was real character there and we faced the challenge head on.
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“We knew Blackpool were very direct and we played on the front foot and caused them problems, especially in the first half-hour and we were in a comfortable position and then the red card came.
“But with the second half, we made clear that Blackpool would throw everything at it.
“They are a very determined side under Mick [McCarthy]. But it was an unacceptable performance.
“We should not have come away with anything other than three points.”
Huddersfield are in danger of slipping down to League One for the first time since 2012, just four years after relegation from the Premier League.