Zenit reportedly tabled a €30m offer for the forward, a bid Benfica were prepared to accept, but the player and his father are reluctant to complete the move.
The Arsenal target is believed to feel a move to Russia would be a step in the wrong direction for his career, and that his long-term prospects are best served by moving to a more prestigious league.
The Zenit offer is thought to be still on the table for Rodrigo and if nothing emerges before Friday's deadline, he may end up moving to the Russian giants.
But Arsenal remain his first-choice, and the report claims Rodrigo believes a strong second half of the season at Benfica could convince Arsene Wenger to snap him up in the summer.
Something for Arsene Wenger to consider - the Russian transfer window does not close until the end of February.
So the Spanish under-21 international may not still be there at the end of the season, should Wenger opt to bide his time and wait until the summer.