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All three players had a year remaining on their contracts, and Thompson revealed Newcastle will receive transfer fees for Flood and Tait, whose destinations are currently unknown.
Their exits rounds off a dismal season for the Falcons, who lie 10th in the Guinness Premiership ahead of Sunday's home game against play-off challengers Leicester.
One report on Saturday claims centre Flood will join Leicester, with fellow back Tait on his way to Sale Sharks.
Flood though, is not available for Sunday's appointment against the Tigers due to an ankle injury that is likely to require surgery and possibly sideline him for England's New Zealand tour next month.
Flood, who has won 17 caps, publicly criticised the circumstances surrounding John Fletcher's departure as Newcastle rugby director in March.
Tait, meanwhile, has collected 23 caps, and scored a try during England's RBS 6 Nations victory over Ireland at Twickenham seven weeks ago.
After Fletcher left Kingston Park, Steve Bates was put in interim charge, but the Falcons have still failed to fire despite fly-half Jonny Wilkinson enjoying a relatively injury-free run.
Thompson said: "Having four of our first-choice back line away with England for so long this year, and the fact that they would miss half of our league games next season, has been a major consideration in the decision to allow both Toby and Mathew to move.
"The impact of four of our back line going out en-masse and then returning at various stages of the season has caused us certain difficulties as we look to get this team to where it needs to be.
"We felt that the situation had to be addressed for the long-term good of the club.
"While we remain committed supporters of England and of English rugby, the interests of Newcastle Falcons remain our foremost concern, and it is with that in mind that we have agreed to these changes."
Flood, Tait, Wilkinson and centre Jamie Noon have all been England regulars this season, but Falcons supporters will still feel mystified at the loss of two such influential figures.
Woods, meanwhile, is a highly-rated back-row prospect who looks a strong bet to gain England Saxons selection for next month's Barclays Churchill Cup.
Thompson added: "All three players still had a year left to run on their contracts, and the club will receive transfer fees for both Toby and Mathew.
"We are currently in advanced negotiations with two Super 14 players, and hope to have some positive news to report on that front in the near future.
"I would like to thank Toby, Ben and Mathew for their contribution during their time at Kingston Park, and wish them well for the future at their respective new clubs."
If Flood heads to Welford Road, it would be a major coup for the Tigers, who look to have missed out on landing Argentina World Cup star Juan Martin Hernandez from French club Stade Francais.
And Tait, should he move to Sale, would undoubtedly fill the full-back vacancy left by Ben Foden's summer departure to Northampton.