Rodriguez, 27, was proposed by Milan to the Russian club as part of a swap deal for attacker Aleksey Miranchuk.
However, the Swiss international told Milan he wants to be the one to choose his next club.
Rodriguez is out of contract in a year and has no intention of being a pawn in the Miranchuk deal.
Milan will have to offer another player – possibly Diego Laxalt – or pay Lokomotiv’s asking price for Miranchuk.
Miranchuk, 24, who can play as a central attacking midfielder or right winger, has scored 16 goals in 32 matches this season.