Chelsea left-back Filipe Luis insists he does not want to leave the club despite struggling to earn a starting place.
Gerard Pique says Barcelona are prepared to lose Xavi in the summer as reports link the midfielder with a move to Qatar.
Southgate, who begins work immediately, succeeds Stuart Pearce, whose contract was not renewed by the Football Association following a disappointing European Championships campaign in June.
Southgate said: "I'm delighted to get this role and extremely excited about the prospect of working with the best and brightest young players in the country.
"I'm a proud Englishman and playing for my country was the pinnacle of my playing career.
"Since retiring as a player I have gained a lot of knowledge and experience of the game both here and abroad, and I'm eager to play my part in preparing players to compete at the highest level.
"Whenever any team steps onto the pitch for England, being successful is of huge importance, but it is also about developing a clear style of play that can allow our talented young players to flourish."