Aliadiere has been a long-term target for Southgate and is not being viewed as a replacement for Mark Viduka, who joined Newcastle on Thursday after rejecting the offer of a contract extension.
Boro's chief executive Keith Lamb said: "We have agreed a fee with Arsenal for Aliadiere and completed a deal with them.
"Everything is also agreed in principle with Jeremie and his transfer will be completed in the next couple of weeks.
"The only reason the deal hasn't already been completed is that the player is on holiday."
Southgate has made no secret of his admiration for the Frenchman, who has been with the Gunners since he was 16.
Lamb said: "We tried to sign Jeremie in January, but were thwarted by Arsenal's injury problems at the time.
"He fits perfectly the sort of player Gareth wants to bring to the club.
"He has pace and energy, and he is young with his best years in front of him."