The Saints have been left short of attacking options with Jay Rodriguez's recovery from a cruciate ligament injury taking longer than expected and Sadio Mane due to leave for the African Nations Cup next month.
Elia, who has played 28 times for Holland, is seen as the perfect man to bolster Koeman's front three as the club look to maintain their excellent start to the Barclays Premier League season.
"He is one of the players on our list to bring in in January - it's still not done but yes, I am interested in the player," Koeman said.
"If he signs for Southampton, I think he will be a good option.
"We need at least one more player up front because we lose Sadio Mane in January for the Africa Cup
We need another player for that position and he is a good option.
"If he does sign for us, it will first be as a loan deal."
Elia, who is known for his blistering pace and direct forward runs, established himself as one of the brightest young talents in Europe at FC Twente and was named Dutch Football Talent of the Year in 2009.
The 27-year-old struggled to fulfil his early promise during subsequent moves to Hamburg and Juventus but has enjoyed more stability at Bremen, where he made 33 league appearances last season.
"He was a Dutch international, he had not such a good period in Germany but I believe in the qualities of the player," Koeman said.
"He is a wing player and we normally play with both a right winger and a left winger.
"We also like in our style of playing fast people who can play one against one with quality."
Southampton may again have to fend off interest in midfielder Morgan Schneiderlin in January.
Tottenham and Arsenal both pursued the Frenchman in the summer but the club stood firm and Koeman insists even a large bid will not prise the 25-year-old away in the transfer window.
He said: "Will we sell Schneiderlin if we receive a very big bid? No, we do not sell Schneiderlin in January."
Source : PA