The 23-year-old has been one of the Bundesliga’s top young midfielders in recent times, starring for Borussia Monchengladbach during a two-year loan spell from Bayer Leverkusen.
His electric form led to a spot on the world champions’ squad in Brazil over the summer, with his most notable contribution coming when given a shock start in the final.
Wenger is desperate to add more steel to his engine room, yet a short-term fix for the latter half of the season is not seen as a genuine option by the Frenchman, who was criticised for his loan move for Kim Kallstrom earlier this year.
Kramer is understood to be a top target, even though snapping him up from Gladbach next month is likely to be complicated.
Leverkusen want their man back at the end of the campaign, while his current side are determined to keep him for the remainder of 2014/15 as they plot to keep their own campaign on track.
The player himself is valued at around £20m-£25m, but a fee would have to be sorted with his parent club, while an arrangement with Gladbach will be needed for them to free him early.
Morgan Schneiderlin of Southampton and another Leverkusen ace in the shape of Lars Bender are alternatives for the Gunners.