Pochettino, who took over at St Mary's in January 2013, has signed a five-year contract with Spurs, who sacked Tim Sherwood at the end of the 2013-14 Premier League season.
The defender, who is part of Roy Hodgson's England squad for this summer's World Cup, wrote on Twitter: "Would like to thank Mauricio for everything he has done for me. Great manager, great guy and has helped me a lot. Wish him all the best."
Shaw has been linked with moves to Manchester United and Chelsea this summer.