Torino have also had a bid turned down for Nelson, who has informed Leicester of his desire to leave the club this summer. The League One side, relegated from the Championship, are said to be holding out for a fee in the region of €12m for the centre-back, who has only a year remaining on his contract.
The 6ft 3in right-footer has already made his wish to leave known to the Leicester hierarchy, with West Ham United and Manchester United also credited with an interest in the former England youth international.
Nelson has previously been capped at Under-20 level for England, and impressed enough during Leicester's difficult recent campaign to attract attention from clubs in three different countries.
Leicester's stance reflects their determination not to lose the academy graduate for a cut-price fee, particularly given his contract situation leaves them with limited time to negotiate before his value diminishes further. The club will hope that competition between Borussia Monchengladbach, Torino and any Premier League interest can help push a deal closer to their valuation, rather than allowing suitors to wait out the final year of his contract in the hope of a cheaper deal or a free transfer next summer.
Nelson made his first-team breakthrough last season, establishing himself as a regular in Leicester's backline despite the club suffering back-to-back relegations from the Premier League to League One.
He is capable of playing at left-back as well as in the centre of defence, a versatility that has added to his appeal for clubs across three different leagues this summer.
Torino's interest is understood to be the most advanced of the group, with the Serie A club having been linked with Nelson since June, though Leicester's asking price has so far proven a sticking point for all parties involved in the chase, including any prospective English suitors weighing up a move.