The 26-year-old defender is in the final year of his contract at the German club and was heavily linked with a move away in the last transfer window.
A move to the Premier League was mooted when Everton and Wolves showed an interest in getting their hands on the centre-back.
However, both the Premier League clubs cooled their interest in Elvedi as the end of the transfer window drew nearer and he eventually stayed out at Gladbach.
Eyes have been on him with regards to a possible January move, but according to German daily Bild (via Fussball Transfers), the defender is now focusing on extending his stay at Borussia Monchengladbach.
It has been claimed that Elvedi is now keen to sign a new deal and commit his future at the club.
As it stands, he can hold talks over a pre-contract with his suitors when the transfer window opens in January.
However, Elvedi is keen to remove that uncertainty and sign on the dotted line of a new contract.