Silva, 39, is high on their list of replacements if Italian Walter Mazzarri departs after just a year in charge at Watford.
The Portuguese boss has impressed in the second half of the season with a solid home record, giving Hull an excellent chance of Premier League survival in the wake of Mike Phelan’s dismal record until January.
Walter Mazzarri has failed to live up to expectations at Watford this termReuters
Should Hull overhaul Swansea and even Crystal Palace, then that could trigger an extension to tie Silva to the club if he pulls off a great escape.
The Hull home crowd have tried their hardest in recent weeks to pull at the heart strings of Silva.
Yet the former Olympiakos boss endured a rocky beginning when thousands of fans boycotted his first match for the FA Cup victory over Swansea in protest at owner Assem Allam.
But KCOM Stadium regulars soon bought into the Silva revolution. Silva said: “The supporters have been fantastic in every game — maybe apart from the first one but we know the reasons for that.
Marco Silva's Hull are staring relegation in the face as it standsRex Features
“Since then, it has been amazing how they have supported our team.
“Their backing for the players has been fantastic, and personally I’ve felt their support for me and that’s been really important.
“They really believe in us and we have a close bond.”
City have not won on the road for nine months but a rare away victory today at Selhurst Park will drag the Eagles into a final day shootout for survival next Sunday — when they travel to Manchester United and Hull host Tottenham.
Walter Mazzarri may be on the brink of the sack at WatfordReuters
Hull can cut the gap with the Eagles to a point with only a second win at Selhurst Park in more than 30 years.
Silva added: “We believe we can go there and win and it’s all about who can keep a calm head and maintain their focus.
“It’s a match that will be decided in the details — set-pieces and things like that. It doesn’t matter about playing well, because at this stage it’s all about winning.
“It’s easy to say there is more pressure on us, but Crystal Palace will feel it as well.”
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