Former Sporting and Olympiakos Piraeus coach Silva, who had signed a contract at Hull until the end of the season, met the club's owners on Wednesday and was expected to quit.
"Marco was offered the opportunity to continue the work he has done in the four-and-a-half months he has been at the KCOM Stadium," Hull said on their website (www.hullcitytigers.com)
"Unfortunately, after considering his future, he has chosen to leave in order to further his career ..."
The club's Vice-Chairman Ehab Allam told Sky Sports: "The way the contract was structured meant there would be one year left on his contract had we stayed in the Premier League - but with relegation unfortunately he had the option to go.
"He's been fantastic for us but I accept he's a better calibre manager than the Championship. I understand why he wants to stay in the Premier League or go to a big European club with the prospect of Champions League football."
The Portuguese was reported in British media to have had talks with Porto on Tuesday, after the club parted company with coach Nuno earlier this week, and has also been linked with the vacant jobs at Premier League Watford and Crystal Palace.
Silva guided Hull to six wins from his 18 league games but they were relegated after a 4-0 loss at Crystal Palace on May 14 and finished with a 7-1 home thrashing by Tottenham Hotspur.
"We'd love him to stay. We're very sad to lose him but I respect why he's taken the decision," added Allam.
"We attained 21 points in 18 games (with Silva) and if we'd had that through the season we'd have been mid-table.
"I can understand why there's so much interest in him. The improvement was there for all to see and if he gets a full season anywhere else I'm sure he'll do a fantastic job."
Hull said the club would "now undertake a thorough and considered recruitment process to appoint a replacement" as they look for their fourth manager in less than 12 months.
(Writing by Ken Ferris; Editing by Alison Williams)
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