The Netherlands international, who has 27 caps for his country, will complete the move when the transfer window re-opens on January 3.
Southampton have the option of making the loan move permanent at the end of the season, while Elia will be available to make his debut against Manchester United on January 11.
Speaking prior to completion of the deal, Saints boss Ronald Koeman spoke of his admiration for the 27-year-old and is confident that Elia can get back to his best after a lacklustre few seasons in Germany.
"If finally the player signs for Southampton I think it's a good option, we need at least one player up front because we will lose Mane in January to the Africa Cup of Nations," he told reporters.
"We need one player up there. It's a good option, he's a Dutch international. He didn't have such a good time in Germany but I believe in the qualities of the player.
"He's a good wing player. We like playing one right winger, one left winger, we like fast players who can play one-on-one. He's one of our options. If the deal is done, it will be first a loan deal."