Gareth Bale has indicated that he felt a need to be the main man at Tottenham, saying he tried to "make everything tick".
Southampton manager Mauricio Pochettino has vowed never to manage a club without total control of transfer policy.
The 22-year-old striker's three-month stint with the Clarets came to an end after Saturday's 4-0 defeat at Blackpool.
"I think he's done well. He's filled in the various positions, which is the job that we wanted him to do," said Burnley boss Eddie Howe.
"He probably could have scored more than the one goal he got, but I think he can be really pleased with his contribution.
"He's been a joy to work with, he's been very good for the squad, and we wish him well going back to Millwall."