The 23-year-old centre-back, a 17-cap Swiss international, is having a fantastic season for Gladbach.
His contract expires in June 2021 and if the Bundesliga side fail to extend it this summer, he may have to be sold.
Elvedi is valued at €28m by Transfermarkt, but due to his contract situation his price could be lower.
And in an exclusive report, le10sport claims both Arsenal and Bayern Munich are showing clear interest.
At 1.89 m (6 ft 2 in) and with excellent technical ability, Elvedi could be an astute buy for either club in the transfer window.
Elvedi is one of the best passers in Germany, so has an ideal profile for both the Gunners and Bayern.
In the Bundesliga and Europa League this season, he has averaged a 92.4% pass completion rate for Gladbach.