Giroud shot to prominence last season, with his 21 goals helping La Paillade seal the first Ligue 1 title in their history.
The Gunners are believed to have agreed a £12 million fee with the French side to sign the 25-year-old.
And Gladbach sporting director Max Eberl admitted it is hard for his side to compete with the North London side, despite qualifying for the Champions League.
"We remain true to our policy that we do not throw our money out the window. The problem in our negotiations is that we are in competition with clubs like Arsenal and top clubs in Germany," he said.
Reports now suggest the Bundesliga side are to turn their attentions to FC Twente front-man Luuk de Jong or John Guidetti of Manchester City.
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